People seem not to understand why hallel is said in shul (by those who say it), and why it is not said by those who don’t. I alluded to this earlier; now let me explain in detail.
Hallel is a mitzvah derabbanan, and the way Chazal instituted for it to be said normally is standing, and without interruption. Like any mitzvah derabbanan we say a bracha before it, asher kid’shonu bemitzvosov vetzivonu. On Pesach night, however, Chazal instituted that it be said sitting, at the seder, and in two separate sections with a long interruption between them. Bedieved one is yotzei if one says it in the normal fashion, but one is not doing the mitzvah as it is supposed to be done.
However there is a three-way machlokes about the brocho. Some rishonim hold that we should say a brocha before starting the first section of hallel, just as we always do. The interruption doesn’t matter, because that is how the mitzvah is supposed to be done this time. Other rishonim hold that we say the brocha twice, once before each section, since the mitzvah tonight is to say it in two sections. And other rishonim hold that since there is this interruption we do not say a brocha at all.
In practice we pasken that sofek brochos lehokel, and therefore we don’t say the brocha. But that means that according to the first two opinions we are missing something that we should be doing. Therefore the minhag arose in Sefarad, and was later adopted by chassidim, to say hallel after maariv, the regular way, even though that is not the way it’s supposed to be done, in order to say the brocha. Then we go home and do it again properly.
However this has a big down side. When we come to say Hallel properly we have already been yotzei the mitzvah, so we cannot now fulfill it properly as it was instituted. It’s the same issue as saying krias shma before davening, without the brochos. We are advised in that case to say it on condition, that if we will get to krias shma in davening before the zman then we now have intention davka not to be yotzei. But with the Pesach hallel there is nothing for such a condition to apply to. There is nothing that will either happen or not happen, on which we can condition our intent to be yotzei. Therefore minhag Ashkenaz is davka not to say this hallel, sacrificing the opportunity to say the brocha in order to be able to be yotzei the mitzvah in the proper manner. Minhag Sefarad makes the opposite sacrifice.
According to this reasoning, if you say the hallel after davening without a brocha you are nimtza kereach mikan umikan. You aren’t doing the mitzvah ketikunah, and you aren’t gaining the brocha. So what have you achieved?