When should one start building one's Sukkah?

Question

Some people start putting their Sukkos up the week of Rosh Hashanah. I understand that people don't want to do things until the last minute. However, in my opinion, the earlier one starts putting his Sukkah up, the less enthusiasm one has for the mitzvah. Why? If one puts his Sukkah up earlier, it appears that it's a burden on him that he wants to get done with as soon as possible.

1. What does halacha say about this?
2. When should one start putting up his Sukkah?

Answer 1

The Ramma both in siman 624 (end of hilchos Yom Kipur) and in the begining of hilchos succah siman 625 says to begin right after Yom Kippur so as to go from mitzvah to mitzvah.

EDIT

The seffer Minhag Yisroel Torah in siman 602 ois 3 brings the following. The Elya Rabba says to be carefull not to build the succah before Yom Kippur because going out to the Succah is like going to exile and if one gets a psak din on Yom Kippur to go to galus, he would have been able to fulfill the gzeira by building the succah, if its already built, he can't use this. He brings from the Shaarei Tshuva however that says building the Succah is not what the Medrash means about a Succah being like galus, sitting in it is. So it would be better to act in accordance with the Levush and build it before Yom kippur so as to have more Zechussim. He then adds on in the from the Birkei Yosef to at least put the walls up before Yom Kippur so as to get the zchus like the Levush and still be yotzeh the Elya Rabba.

The seffer then goes on to quote some Chassidish sfarim, how some would start building the succah on the 6th day of the aserres yomei tshuva because the slichos of that day mentions sitting under Hashem's shade. Also another minhag was to start preparing the decorations before Yom Kippur, but not the actual Succah.

Answer 2

Based on this article:

When should one build the Sukkah?

1. The pious are careful to build to begin building the Sukkah on Motzei Yom Kippur and finish it the next day in order to go from one mitzvah (Yom Kippur) to another (Sukkot).

2. If there is a shabbat between Yom Kippur and Sukkot some poskim say that there is an obligation to build it before that shabbat. Avnei Nezer 459 writes that this obligation isn't just because of zrizin makdimin limitzvot and therefore it is an obligation. However, the Minchat Elazar 4:55 writes that this is not an obligation but just the preferred option, and this is the conclusion of Yalkut Yosef page 118 and Chazon Ovadia Sukkot page 99.

3. Preferably, one should build it for himself, but if this isn't possible he should appoint an agent to do it for him and strive to at least take part in the building in some form even if this means missing out on more learning time.

4. If a Sukkah was set up with the S'chach from before 30 days before sukkot and it wasn't made for the purpose of the mitzvah of Sukkot one must change or improve one thing (such as putting down S'chach) of an area a Tefach by a Tefach or the whole width even if it's of minimal length (or vice versa). However, if it was made within 30 days of Sukkot or was made for the express purpose of Sukkot it's fit even without any change. However, if the Sukkah was originally built for Sukkot and stands unchanged year to year it requires some change the each year. This is only if the S'chach was put in advance but if just the walls were putup in advance and not the S'chach there's no issue.

From these rules, I infer that it is preferable to start after Yom Kippur, but there doesn't seem to be a problem starting before. One year, my shul put up the succah before Tish'a B'Av. Yes, I thought it was bizarre, but they explained that the shul's maintenance supervisor, who constructed the succah each year, was going a long vacation, and he wasn't planning on returning until just before Succot, not leaving enough time to do it. (Separate question, perhaps, re the maintenance guy wasn't Jewish...)

My understanding is that the succah isn't complete (i.e., it's not an official Succah) without the schach. So, based on rule 4, above, I believe that if you put the scach on after Yom Kippur, you're fulfilling the technical terms of "building a succah", no??