Just brainstorming here. Buchanan basically kicked the can down the road on how to handle states seceding. That made the Civil War worse by allowing federal authority and federal arms to be seized by the secessionists. It could have been worse, though.
What if Buchanon died shortly before or after the 1860 election? That would leave his Vice President, John C. Breckinridge, in charge for the remainder of Buchanan's term--from November 1860 to March 1861. That would stir things up. Historically, Breckinridge was very pro-South, to the extent that he ended up serving in the Confederate army, though he did attempt to find a compromise that would allow the seceding states to stay in the Union before war broke out.
Not sure how that would cut in terms of secession. I'm also not sure how the North would react to having a pro-slavery president from a slave state (Kentucky) in charge for several months while the nation was in political turmoil. They wouldn't have enough people in the then current Senate to impeach him. What would happen, for example, if he quietly allowed all federal property in the seceding states to be seized by the states, and/or withdrew federal troops from seceding and border states as a "gesture of reconciliation", leaving the states in charge of the forts and arsenals?
That could end up with some very nasty situations in the likes of Missouri, with mini-civil wars breaking out in several border states, and maybe even mutinies among pro-North military leaders. That could get extremely twisted, with pro-Confederate elements of the military claiming to be supporting constitutional authority while they try to suppress pro-Union rebellions in the military.
If you want to get really nasty, have Breckinridge's actions inadvertently or deliberately push the more radical elements in the north into open armed rebellion against his authority. Maybe he does the withdrawal and base turnover, then adds in giving the seceding and border states "their" share of the US Navy and Treasury, maybe even control of "their share" of the territories that hadn't become states yet. I'm pretty sure that would spark a rebellion. Then the federal government spends the next couple of months trying to suppress the rebellion, with pro-Confederate forces happily joining in to "support the constitutional authority of the president."
That's an interesting--actually rather nightmarish--question in itself, but it isn't "What if the south didn't secede?" Could Breckinridge have calmed the fire-eaters down in some way, by offering them the concessions mentioned and maybe others? That would be difficult because a temporary presidency wouldn't have much to offer beyond federal property that couldn't be rolled back when Lincoln took power.
Maybe we take the northern rebellion a step further. How would president-elect Lincoln react to Breckinridge attempts to make secession or at least the emasculation of federal power a fait accompli? It would be difficult to say nothing as the current president deliberately undercut future presidents' power. In the climate of an open armed rebellion, though, he would have to be very careful not to say something that could be construed as inciting rebellion. Worse case scenario: we end up with the current president of the United States trying to arrest the president-elect as a traitor.
Bottom line, the south blunders into a situation where they temporarily control the levers of the federal government and maybe into a kind of near-coup situation, with the north being forced to rebel against the current federal government in order to keep the president-elect from being arrested for treason. At that point, I could see the two sides each claiming to be the legitimate federal government, with pro-North military men rebelling against the Breckinridge regime and forming their own army to protect Lincoln. The army tears itself apart. We have a soon-to-be president in armed rebellion against the current president. At that point, I have no idea what happens.
Maybe Breckinridge declares the 1860 elections void given the 'rebellion' and tries to hold new elections, which tears the country apart even more completely. The south doesn't secede so much as it tries to retain control of the federal government. Maybe the Breckinridge administration even declares rebelling Northern states, or maybe just the Republican Party as being "in rebellion" and refuses to seat their incoming Senators and Representatives. We end up with the south in control of the symbols of federal power, and probably in control of most of the border states and some territories, but otherwise with pretty much the same division of the country.
And that goes in some very dark directions. I'm not sure Breckinridge would go down those roads. While he was pro-slavery, I'm not sure he would be willing to be as dishonorable as he would have to be to get the country in that kind of situation. In any case, this does illustrate that secession wasn't the worst situation the country could have found itself in over slavery.