The real fun and games (butterflies permitting) would be in 1824.
If there is no 12th Amendment, then the HoR gets to choose from the first five candidates, not just the first three. So Henry Clay would not be excluded from the contingent election.
Also, the possibility arises of some of the candidates "pairing off". Might Adams and Clay make their "bargain" before November rather than after it, agreeing that the electors of each shd give their second votes to the other? If Jackson and Calhoun did likewise, this could produce a situation where the Crawford Electors have a "casting vote" as their second choices could decide who became POTUS. This, I suspect, would quickly produce a 12th Amendment of some kind, though not necessarily identical with OTL's.
All this, of course, assumes there has not been a 12th Amendment already, but of course there might be. To briefly remount an old hobby-horse, in 1820 an Amendment to choose Presidential Electors in districts came within six votes of passage in the HoR. TTL, might someone add a rider to it for "designation" of Presidential and Vice-Presidential candidates, and might this attract the few extra votes required?