How do Chassidim and litvaks who hold Rabbeinu Tam's view of when nightfall is (later than most people's) blow shofar before that time after neilah?
Mishna B'rura 623:12 says one may blow shofar after n'ila even during ben hash'mashos (i.e. before sure nightfall, as long as it's safek layla, possibly night), since, as he says there, "it's a rabbinical prohibition for the sake of a mitzva". Since you're asking about Rabenu Tam's view, you may prefer a more hasidic source: Shulchan Aruch Harav :11 permits it also ("since this sounding is not a labor at all and doesn't even have a complete prohibition… and was prohibited on Shabas and holidays only because of 'weekday activities'"). See also Taame Haminhagim 773 (which does not address your question exactly but is interesting).