1) First you should approach a knowledgable Jew who is also someone you can trust enough to shmooze it over with.
This step is not asking for an actual "psak halachah" from a 2nd Rav. This is simply you getting an education on the Torah matter. Yes, you should not pretend ignorance when other information comes up. Rather, you should investigate it.
It is important here to mention a point. In general, No Jew "MUST" ask a Rabbi something. You are supposed to know the Torah yourself. You only ask a Rabbi when you have a doubt.(Devarim 17:8 "If a matter of judgment be hidden from you...") Therefore, nothing stops you from learning Torah! So, go ask 10 people their opinion. Only, do not ask it as a request from a Rav for a psak. You may ask a Rav (or many) for a theoretical opinion, without being bound. (Rav Shlomo Zalman Auerbach)
Also, did you in fact intend to go to the first Rabbi for a real psak? Did you ask with intent that his decision be legally binding upon you? Did the Rabbi understand that this was what you were doing as well and issue a psak in his intention? OR was it understood that the conversation was initially about serious advice and did not take the form of a formal psak Halachah? If the latter, then you are not bound at all.
2) Three Possibilities
Decide, (once you have fully understood the new info from the conversation with the Dayan, and how it applies to your situation) if your 1st Rabbi:
A) made an error that violates halachah, or goes against the "overwhelming" majority or mainstream accepted practice. OR his decision was based on a wrong presentation of the facts.
B) He didn't make an outright error, but you now believe strongly that you can present the case in a way that the 1st Rabbi will admit the whole thing should be reversed. OR you found enough mainstream opinions that say against him and you feel he will change his mind due to those opinions.
C) It is a question of basically equal balance and your Rabbi's opinion is perfectly valid, even if there are other conflicting opinions.
3) Three Outcomes
If it is A, then you should disregard the 1st Rabbi's decision.(or at least you have the right to disregard) It is invalid. You should still respect him, and may even ask him questions in the future. After all, we are all human. :) (Tosfos to Avodah Zara, 7a; Bechoros 28b)
If it is B, then you may bring the case back to him and ask him to re-examine the question in light of the new information. He is allowed to change his mind. (Shach to Yoreh Deah 242:58; Rosh Sanhedrin 4:6) Some say he may retract even if he doesn't believe he is proven to be in the wrong.(Levush 242:31, Aruch HaShulchan 242:60)
If it is C, then his decision stands. Some authorities rule that even if he changes his mind, it has no effect.(Shach) Others rule that you may ask him to change his mind and he may do so.(Levush, Aruch HaShulchan)
Of course, if you discover that the 1st Rabbi was right all along, then this whole question is moot. :)
However, in the first place, one should generally never approach a Rav for a psak. Rather, ask him to teach you Torah, give you "eytzah" advice, talk in learning, share his opinion on the subject, etc.
If and when you are freely ready to give up your will and want to accept his opinion as obligation, because you don't know the answer yourself and have no choice, then and only then, ask for a real Halachic psak. Then it is binding. Otherwise it is not; and all of the above wouldn't matter.
If you already asked for a psak, you are anyway only obligated to follow it for a one time event.(Rema 242:31) Example: If he forbids you to make ice cubes on Shabbos, then you can't make them this Shabbos. Next week you can ask another Rabbi. OR go learn the issue and pick a path for yourself. :)
If you did accept a psak, from a Rabbi, then there are two possible reasons why you must now listen to the psak. A) Asking a second Rabbi would be an insult to the honor of the first Rabbi (and by extension dishonoring the Halachic system).(Ran, Rosh) OR B) Once you knowingly ask a Rav for a psak, it is as if you vowed to forbid the object the Rabbi forbids, upon your own soul. "Shavya A'nafshey C'chatichah D'issurah". (Raavad, Ramban, Rashba, Shach, Gra)
So, if the psak was stringent, and now you seek to have a second Rabbi be lenient, then the self imposed vow applies. The second Rabbi cannot reverse it. But if the psak was lenient and now you want to ask a second Rabbi to be strict, its fine.
(BTW, according to this, if a psak is neither lenient nor stringent (ex: Do you need to repeat Shmoneh Esrei?) then the reasoning doesn't apply, and you may ask another Rabbi too.) Finding out if your question is stringent or lenient can be quite a trick. :) Multiple and equal stringencies and leniencies in the same question usually cancel out so that the question is considered neither stringent nor lenient.
But, if we are worried for the honor of the 1st Rabbi, then it doesn't matter if he was strict or lenient. A second Rabbi may not reverse it either way.
Some are worried for both opinions. (Rosh, Rema)
Either way, if you do ask a second Rabbi for a psak, you must explain to him that you already asked the first Rabbi and you are seeking his help to convince the first Rabbi to retract. (Tosfos to Avodah Zara, 7a)
I hope this helps. :)