How does this happen? IOTL, Buchanan beat Fremont by 12% in the popular vote. Lets say that Franklin Pierce is renominated in a bitter convention fight, and Fremont manages to carry Illinois (4% Buchanan margin IOTL) and Pennsylvania (18% Buchanan margin IOTL, but Buchanan was from Pennsylvania, in addition to the states he carried, and gets an electoral college majority, even if he doesn't win the popular vote.
Somehow we get Fremont elected President in 1856. What happens after that?
In 1856, Chief Justice Taney wrote to his son-in-law, predicting that either Frémont or Fillmore would win. Taney regarded both of them as Northern men, hostile to the South. He thought the South ought to "separate", but that either the Whig or Republican administration would build up a faction of patronage stooges in the South that would prevent it.
If Taney felt that way, the real Southern Fire-Eaters would be howling for immediate secession. But would they succeed?
Southern voting was closer on that issue in 1860 than has been generally realized, and the newly organized Republicans were much weaker in 1856, posing less of a threat. (Republicans held only 90 of 237 seats in the 1857-1859 House OTL; if Frémont runs stronger, they might win 15 more, but would still be a minority. They held only 20 seats of 62 in the Senate.)
The
Dred Scott decision had not emboldened Southerners. John Brown's attack at Harpers Ferry had not frightened them.
And one other point: perhaps minor, perhaps key. Matches were introduced around this time, but these early matches were not "safety matches". They had an unfortunate tendency to "sweat" material from their incendiary tips, and to ignite spontaneously. This often resulted in seemingly mysterious house fires. Texas had a particularly hot, dry summer in 1860, and there were many such fires. During this period, the Fire-Eater press printed many stories about "abolition fiends" circulating through the South, inciting slaves to murder and arson, including many reports of such "fiends" caught with supplies of poison and incendiaries (always in some remote area of another state). The wave of mysterious fires plaguing Texas boosted this campaign substantially.
This was important, because a major reason for Southern secessionism was physical fear of slave rebellion, especially among whites who lived in slave-majority areas. For the moment, no slave rebellion could truly succeed, because the local whites would be reinforced by whites from elsewhere and ultimately from across the country. But if the Federal government was controlled by "abolition fiends", white control would be attacked from below
and above, and could collapse. This fear was present in 1856, but due to Harpers Ferry and the Texas house fire epidemic, was substantially more acute in 1860.
Here is another point. Even South Carolina was not united for immediate unilateral secession in 1860. There was a faction which argued that South Carolina should propose secession, and invite the other slave states to a convention. That convention would monitor the Lincoln Administration's actions, and when those actions justified it, issue a call to secession on behalf of all the Southern states. This would avoid the risk of South Carolina declaring secession, and finding itself alone. But sometime in the month after the election, there was a ceremony to mark completion of a new railroad from Georgia to South Carolina. This ceremony was attended by prominent men from both states, and the Georgians assured the South Carolinians that Georgia would follow South Carolina's lead on secession. Thus reassured, South Carolina proceeded with immediate secession.
In 1856, secession sentiment would be weaker, for reasons given above, and the delayed-action faction would be stronger. Frémont was not Lincoln, but he was no fool either. He would see his weak position and act cautiously. He would avoid giving the Fire-Eaters or the slave-states convention any excuses to call for secession. And after a few months in office, the panic would have died down, leaving the secessionists high and dry.
To be sure, it all depends on what Frémont would do in office. Frémont himself was a former Democrat, and the son-in-law of Missouri Democrat former Senator Thomas Hart Benton. If he attempted some radical anti-slavery action, that could set off secession. But he would be very limited in what he could do, given control of Congress by Democrats and other slave-state members. Any appointment he made would have to be acceptable to a lot of Democrats. If it was clear that he was hamstrung by the limitations of Presidential power, that would be a strong argument against secession: "What's the big deal? We've had an anti-slavery President and nothing happened. Why ask for trouble?"
Taney would still issue
Dred Scott, perhaps making it even stronger to constrain Frémont. OTOH, Frémont would support the Free-Soil settlers in Kansas, which would probably achieve statehood in 1858.