What if Jackson wins in 1824? John Quincy Adams goes to the House of Representatives in 1826 and Jackson serves his two terms, Nullification Crisis as usual right near the end of his term, Van Buren wins In 1832 over clay the John Quincy Adams sees the problem of Southern slaveocracy growth and runs in 1836 because clay has already lost a couple times. He wins because the financial Panic is moved ahead some since Jackson was in earlier.
So, instead of being when Congress to continue fighting against the rule that they couldn't discuss slavery, Adams is President. Yes he's older, but he's pretty spry. He goes on against the south in the Amistad case, with his secretary of state Daniel Webster being asked to argue for freedom in it. He also puts the dampers on any Prospect of Texas entering the Union because it would be a slaveholding state.
You could possibly see the south in enough of an uproar that they secede in the middle 1840s with one of their goals to Annex Texas once they secede. The fire eaters in our timeline were already very active by 1850 in some states and could be more so here as they see presidents actively going against their power.
The thing is that you need to figure out which states would secede. In my timeline on Webster being vice president and then becoming president after Harrison dies, I have William Seward becoming president in 1852 and Missouri is run by someone who supports the session and they wind up seceding but Tennessee doesn't because of who is in control of that state. You could have a few different scenarios based on who is running what.
Of course, Adams died in 1848 in our time line, I don't know if he would want to serve two terms if he is elected in 1836 but if not maybe Webster runs and wins in 1840. However, because Adams did seem to see the dangers of the growing slave power in Congress in our timeline and fought hard to opposed the gag order, so to speak, on the discussion of slavery, it is possible that he could sense something and decide that he needed to try again.