Concerning the three steps at the end of Shmoneh Esrei, there are three opinions mentioned in the Shaarei Aharon on shulchan Aruch.

1. The Orchos Chaim (#24) says to move the right foot back first a small step. Then with the left, a large step. Afterwards, with the right a small step to make it parallel to the left.

2. The Shulchan Aruch says to start with the left, the size of these steps should be at least large enough to place the thumb toe next to the heel, the simple reading implies that one should start with the left, then move the right so the right toe is next to the left heel, and afterwards move the left again make it even with the right. This is what the Levush writes and ends off by saying this is the humble way to do it. As brought in Machatzis Hashekel.

3. The Chazzon Ish says to take three large steps and afterwards to make them equal with each other. He seems to explain the Shulchan Aruch this way.

Untill here is from the Shaarei Aharon.

As far as the half steps you mention, I remember reading about this but I don't recall where. Apparently this is the minhag of many Hasidic Rebbes. The explenation I saw was based on the Beis Yoseif who says one takes three steps backwards, then after the three steps forwards, he can go on his way, after a completion of six steps. Taking this idea of six steps and adding it with the idea that making your feet parallel counts as a full step even though only a half step was taken would mean that splitting the three steps backwards into six half steps is actually counted as six steps and one may now go on his way without going forward. This would avoid the confusion of what one should do after taking three steps back and now needs to go somewhere, say the bathroom. What should he do? Go now and take three steps forward when he gets back, or take three steps forward now? For instance, the Kaf HaChaim held like the first option, Rabbi Reuvein Margolis in his Nefesh Chaya held like the latter. The practice of these Rebbes should avoid this issue, according to the seffer that I can't recall the name of.

Regarding the steps before Shmoneh Esrei, there seems to be confusion involved, as there is no reason to take steps backwards. There is an opinion brought in the Ramma in the name of the Rokeach in siman 95 to go three steps forwards, as a means of 'coming close and hagasha (a word associated with many people praying in the Torah) to fulfill your duty'. The Mishna Berurah there says explicitly one need not go backwards in order to do this act, but the minhag of the world is in fact to go back.

The Kitzur Shulchan Aruch mentions to go back three steps when you say Tehilos l'Kel elyon etc. Apparently, he feels in this way a person is considered being back there and is now actually moving to a new place. Perhaps even he agrees that going back right before Shmoneh Esrei is not a good option.

Noone I saw describes these three steps. One can imagine they would be the same, but care was taken to describe a practice mentioned from the gemara, that is going back afterwards, whereas an idea from the rishonim, to go forward is not as worthy to go into such details. Especially if the Levush doesn't mention it and some take that as meaning he doesn't agree with it. Also, the logic about the six little steps from the unnamed seffer would not apply, as here, there is no idea of six steps, only three.

Concerning the three steps at the end of Shmoneh Esrei, there are three opinions mentioned in the Shaarei Aharon on shulchan Aruch.

1. The Orchos Chaim (#24) says to move the right foot back first a small step. Then with the left, a large step. Afterwards, with the right a small step to make it parallel to the left.

2. The Shulchan Aruch says to start with the left, the size of these steps should be at least large enough to place the thumb toe next to the heel, the simple reading implies that one should start with the left, then move the right so the right toe is next to the left heel, and afterwards move the left again make it even with the right. This is what the Levush writes and ends off by saying this is the humble way to do it. As brought in Machatzis Hashekel.

3. The Chazzon Ish says to take three large steps and afterwards to make them equal with each other. He seems to explain the Shulchan Aruch this way.

Untill here is from the Shaarei Aharon.

As far as the half steps you mention, I remember reading about this but I don't recall where. Apparently this is the minhag of many Hasidic Rebbes. The explenation I saw was based on the Beis Yoseif who says one takes three steps backwards, then after the three steps forwards, he can go on his way, after a completion of six steps. Taking this idea of six steps and adding it with the idea that making your feet parallel counts as a full step even though only a half step was taken would mean that splitting the three steps backwards into six half steps is actually counted as six steps and one may now go on his way without going forward. This would avoid the confusion of what one should do after taking three steps back and now needs to go somewhere, say the bathroom. What should he do? Go now and take three steps forward when he gets back, or take three steps forward now? For instance, the Kaf HaChaim held like the first option, Rabbi Reuvein Margolis in his Nefesh Chaya held like the latter. The practice of these Rebbes should avoid this issue, according to the seffer that I can't recall the name of.

The Igros Moshe Orach Chaim chelek 4 siman 122 #2 also discusses the three steps according to the Gra and what the majority of people do, and mentions that being that the personal practice of the Gra was not mentioned in his siddur or Maaseh Rav, 'this implies there is no preference how one steps'.

Regarding the steps before Shmoneh Esrei, there seems to be confusion involved, as there is no reason to take steps backwards. There is an opinion brought in the Ramma in the name of the Rokeach in siman 95 to go three steps forwards, as a means of 'coming close and hagasha (a word associated with many people praying in the Torah) to fulfill your duty'. The Mishna Berurah there says explicitly one need not go backwards in order to do this act, but the minhag of the world is in fact to go back.

The Kitzur Shulchan Aruch mentions to go back three steps when you say Tehilos l'Kel elyon etc. Apparently, he feels in this way a person is considered being back there and is now actually moving to a new place. Perhaps even he agrees that going back right before Shmoneh Esrei is not a good option.

Noone I saw describes these three steps. One can imagine they would be the same, but care was taken to describe a practice mentioned from the gemara, that is going back afterwards, whereas an idea from the rishonim, to go forward is not as worthy to go into such details. Especially if the Levush doesn't mention it and some take that as meaning he doesn't agree with it. Also, the logic about the six little steps from the unnamed seffer would not apply, as here, there is no idea of six steps, only three.

There are three opinions mentioned in the Shaarei Aharon on shulchan Aruch.

1. The Orchos Chaim (#24) says to move the right foot back first a small step. Then with the left, a large step. Afterwards, with the right a small step to make it parallel to the left.

2. The Shulchan Aruch says to start with the left, the size of these steps should be at least large enough to place the thumb toe next to the heel, the simple reading implies that one should start with the left, then move the right so the right toe is next to the left heel, and afterwards move the left again make it even with the right. This is what the Levush writes and ends off by saying this is the humble way to do it. As brought in Machatzis Hashekel.

3. The Chazzon Ish says to take three large steps and afterwards to make them equal with each other. He seems to explain the Shulchan Aruch this way.

Untill here is from the Shaarei Aharon.

As far as the half steps you mention, I remember reading about this but I don't recall where. Apparently this is the minhag of many Hasidic Rebbes. The explenation I saw was based on the Beis Yoseif who says one takes three steps backwards, then after the three steps forwards, he can go on his way, after a completion of six steps. Taking this idea of six steps and adding it with the idea that making your feet parallel counts as a full step even though only a half step was taken would mean that splitting the three steps backwards into six half steps is actually counted as six steps and one may now go on his way without going forward. This would avoid the confusion of what one should do after taking three steps back and now needs to go somewhere, say the bathroom. What should he do? Go now and take three steps forward when he gets back, or take three steps forward now? For instance, the Kaf HaChaim held like the first option, Rabbi Reuvein Margolis in his Nefesh Chaya held like the latter. The practice of these Rebbes should avoid this issue, according to the seffer that I can't recall the name of.

Concerning the three steps at the end of Shmoneh Esrei, there are three opinions mentioned in the Shaarei Aharon on shulchan Aruch.

1. The Orchos Chaim (#24) says to move the right foot back first a small step. Then with the left, a large step. Afterwards, with the right a small step to make it parallel to the left.

2. The Shulchan Aruch says to start with the left, the size of these steps should be at least large enough to place the thumb toe next to the heel, the simple reading implies that one should start with the left, then move the right so the right toe is next to the left heel, and afterwards move the left again make it even with the right. This is what the Levush writes and ends off by saying this is the humble way to do it. As brought in Machatzis Hashekel.

3. The Chazzon Ish says to take three large steps and afterwards to make them equal with each other. He seems to explain the Shulchan Aruch this way.

Untill here is from the Shaarei Aharon.

As far as the half steps you mention, I remember reading about this but I don't recall where. Apparently this is the minhag of many Hasidic Rebbes. The explenation I saw was based on the Beis Yoseif who says one takes three steps backwards, then after the three steps forwards, he can go on his way, after a completion of six steps. Taking this idea of six steps and adding it with the idea that making your feet parallel counts as a full step even though only a half step was taken would mean that splitting the three steps backwards into six half steps is actually counted as six steps and one may now go on his way without going forward. This would avoid the confusion of what one should do after taking three steps back and now needs to go somewhere, say the bathroom. What should he do? Go now and take three steps forward when he gets back, or take three steps forward now? For instance, the Kaf HaChaim held like the first option, Rabbi Reuvein Margolis in his Nefesh Chaya held like the latter. The practice of these Rebbes should avoid this issue, according to the seffer that I can't recall the name of.

Regarding the steps before Shmoneh Esrei, there seems to be confusion involved, as there is no reason to take steps backwards. There is an opinion brought in the Ramma in the name of the Rokeach in siman 95 to go three steps forwards, as a means of 'coming close and hagasha (a word associated with many people praying in the Torah) to fulfill your duty'. The Mishna Berurah there says explicitly one need not go backwards in order to do this act, but the minhag of the world is in fact to go back.

The Kitzur Shulchan Aruch mentions to go back three steps when you say Tehilos l'Kel elyon etc. Apparently, he feels in this way a person is considered being back there and is now actually moving to a new place. Perhaps even he agrees that going back right before Shmoneh Esrei is not a good option.

Noone I saw describes these three steps. One can imagine they would be the same, but care was taken to describe a practice mentioned from the gemara, that is going back afterwards, whereas an idea from the rishonim, to go forward is not as worthy to go into such details. Especially if the Levush doesn't mention it and some take that as meaning he doesn't agree with it. Also, the logic about the six little steps from the unnamed seffer would not apply, as here, there is no idea of six steps, only three.