Some form of economic populism might be the most likely scenario. In 1935 Huey Long’s Share Our Wealth Society was very popular in the Deep South states but more thinly spread out in the rest of America. So there was the potential for a regional division there. If Long had lived it’s possible that some form of Share Our Wealth third party could have won the Deep South in the 1936 presidential election & maybe had a spoiler effect in other parts of the country.
If the election led to an unclear winner then it could potentially lead to a larger crisis with at least the Deep South states at odds with the rest of the country. Like UnaiB said that would be a bit similar to what I think happens in Kaiserreich, even if it was not so severe as all that.
Another possibility might have been after the 1896 election, where the Panic of 1893 was still being felt and political divisions got severe. America was pretty starkly divided over the free-silver campaign of William Jennings Bryan, I think that some commentators at the time even said that it was the greatest crisis since 1860 (though that might have also just been alarmist rhetoric). The southern states all went strongly for Bryan & not just because he was the Democratic candidate, the free-silver issue had become very popular there, as well as in the western states.
If the election were closer it might have gotten ugly, with a few of the closest states like Indiana and Kentucky having their electoral results challenged by whichever side lost a particular state. Things could get so hotly contested that it could be similar to 1876 and if Bryan supporters began to claim that he was being cheated out of the election then some in the South and West might start talking about violence.
In any case I think that no matter what the initial cause, if the southern states attempted to secede from the US it is pretty unavoidable that white southerners would start invoking the memory of the CSA as a kind of distinct identity, a kind of southern nationalism, to help support their cause. Both Free-Silver and Share Our Wealth occurred at the height of segregation, so black southerners had little say in those movements and there likely would not be concern about appealing to them.