Secession was not really inevitable upon Lincoln's election, it seems like secession was a political ploy which caught on a few years before Lincoln's election.
Let's say the situation is less radicalized, how does Lincoln's presidency go without a southern secession.
One doesn't need to change the whole climate. Immediate secession went through in part because of an incident which could have not happened.
In late 1860, after the election but before the South Carolina convention met, there was a ceremony for the completion of a railroad linking South Carolina and Georgia. It was attended by many notables of both states. At the ceremony, many of the Georgians assured the South Carolinians that if South Carolina declared secession, Georgia would follow.
This had a major effect at the convention. There were two faction there - those who wanted to declare secession at once, unilaterally; and those who wanted to assure joint action with other southern states before doing anything dangerous. The latter group were called Cooperationists. Their plan was to summon a general convention of the slave-holding states, and get all of them to agree to declare secession together.
This would insure that South Carolina wouldn't end up leading the charge by itself. There were enough doubts among the delegates that the Cooperationist plan might have been adopted.
But the Fire-Eaters brushed aside these doubts. They cited the pledges made by Georgians at that ceremony as evidence that other states would follow.
Now, suppose that ceremony hadn't taken place, or been disrupted by bad weather or something. The convention decides against immediate secession, and calls for a southern convention. The other Deep South states follow suit, and are joined by upper south states as well. (I'm not sure if Missouri, Kentucky, or Maryland would send delegates; I'm sure Delaware would not.) The convention is held in New Orleans in March 1861, after Lincoln's inauguration.
The delegates are a mix of outright secessionists, conditional Unionists, and even a few hard Unionists, there to do what they can to obstruct the secessionists.
The convention issues a manifesto, demanding adherence to the whole pro-slavery agenda: slavery to be allowed in all territories, strict enforcement of the Fugitive Slave Act, a Constitutional amendment explicitly protecting slavery, admission of Kansas as a slave state under the Lecompton constitution, suppression of abolitionist agitation.
This is too much for almost any northerner, except some Doughface Democrats. Lincoln does not even reply directly, leaving that up to newspapers and other political figures. The rejection is vehement, which both empowers and weakens the Fire-Eaters. OT1H, they argue that all the North is hostile, rejecting these just demands. OTOH, moderates argue that the Fire-Eaters are making enemies of all the North with these extreme demands.
There is a period of debate lasting some months. But as the debate continues, passions cool and the sense of emergency fades. There are no mass uprisings by slaves, and no apparent effort by Lincoln to insinuate abolitionist agents into the South.
Lincoln fills a lot of Federal appointments in the South with respectable old Whigs. While OT1H, No True Southerner wants to be associated with the Black Republicans, OTOH
somebody has to handle the mail and collect the customs - and secession hasn't actually been declared yet. This cools the secession fever further.
Lincoln appoints a Republican governor for Kansas Territory, and a military commander who is
not a pro-southern partisan. Kansas settlers vote on statehood again, and endorse the free-soil Topeka constitution by over 80%. U.S. troops block attempted interference by Border Ruffians from Missouri.
When Congress meets in December 1861, the situation is down to a simmer. The first issues taken up are the Homestead Act and the admission of Kansas. Both pass fairly easily.
A somewhat knottier issue is the two vacancies on the Supreme Court, left by the deaths of Justices McLean and Daniel. McLean is replaced by fellow Ohioan Noah Swayne, whom he had recommended. But Daniel was a Virginian, and there was a rule that Justices be drawn from each Federal judicial circuit. Virginia was in the Fourth Circuit, with Maryland and Delaware. Lincoln sweated over this, and ...
I draw a blank. I can't think of anyone who would be a plausible SCotUS nominee from the Fourth Circuit by Lincoln.
Maybe Reverdy Johnson of Maryland.
Another Republican goal was an increased tariff. This was contentious, and after a long dispute only a very modest increase was enacted.
Lincoln proposed the construction of a railroad to California, starting at the Iowa border. This too was disputed, but with the support of northern Democrats a railroad bill passed in the final session, in January 1863.
In foreign policy Lincoln faced only one serious challenge: the French incursion into Mexico...
I'm flailing here. The fugitive slave issue would be red-hot, and I can't even guess what would happen.
But I'm pretty sure that if secession didn't happen immediately in 1860-61, it would never happen at all. Once that is accepted, the Fire-Eaters would steadily lose ground. They might be replaced by a "Die-Hard" movement determined to preserve slavery in the South as long as possible, but not by picking fights with Northerners or trying to expand it.
A big question is what happens to the ex-Whigs of the Border States and Upper South. I don't see how they can explicitly join the Republican Party; but there is no reason for a Unionist label to serve as a bridge.