1. Conquest and Occupation
Like the OTL Civil War and Reconstruction, but with the war interrupted by a temporary "Confederate Victory" truce that the Union later repudiates.
2. Successful Crittenden Compromise
War is delayed for one reason or another. There's difficulties organizing a viable Confederate nation without the four states that OTL seceded over the decision to compel the seceding states back into the Union (VA, NC, TN, and AK) and without the war effort as a unifying principle. And there's also more time for the Union to negotiate concessions for renouncing secession. It's plausible that the CSA in such a scenario would accept reintegration after a few years in exchange for Union concessions.
3. Sudden peaceful reintegration
Several years down the road, the Confederate government is faced with a crisis it can't effectively respond to, and eventually in desperation turns to the Union for aid. The Union's price is reintegration, probably with concessions similar to a Crittenden Compromise scenario to address Confederate reservations about reintegration and to provide a measure of face-saving. It'd have to be a very extreme crisis, or Confederate national identity would have to have become very weak, for reintegration to be the least-bad solution
3a. Major slave revolt in the Deep South. This would probably require major organized outside support to supply and arm the rebels in order to become an existential threat to the Confederacy.
3b. Populist rebellion by poor, landless whites, akin to the 1910 Mexican Revolution.
3c. CSA somehow gets into a war with a European power.
3d. CSA unable to pay its national debt. Faces some combination of military compulsion by European creditors and the prospect of a complete financial collapse due to cascading bankruptcy by domestic creditors. Turns to the Union for a bailout.
4. Gradual peaceful reintegration
The CSA would have a very large land border with and share a network of navigable rivers with another country that is its logical primary trading partner. This country would also share a common language, a similar culture, and similar political traditions. It's only natural that barring periods of active hostilities, the distinction between the two countries would blur.
It'd be pretty natural to establish free trade. Heck, I'm not sure the governments at the time could maintain peacetime trade barriers with each other even if they wanted to. Likewise, there would almost certainly be very loose controls (if any controls at all) on immigration and tourism between the USA and CSA. Next would come at least an informal military alliance, then joint infrastructure projects along the border, then regulatory harmonization programs, and so on. As long-distance transportation becomes easier, it'd become more and more common for people to move between the USA and CSA for casual reasons, and it'd be pretty common for sons of well-off Confederate families to attend prestigious schools in the North.
It'd certainly be plausible for them to persist as separate countries despite close cultural, diplomatic, and economic ties, as per the USA and Canada OTL, but it's also plausible that eventually reunification would become a more attractive option than continued formal independence, especially since unlike Canada, the CSA's residents would consider themselves "American" (just a different flavor of American than the USA), and the CSA wouldn't have strong political ties to Britain.
5. Piecemeal reintegration
In this scenario, individual states would secede from the CSA and rejoin the USA. If the open borders and customs union policies posited in Scenario 4 don't happen, many of the border states might decide that they'd rather be on the Union side of the trade barriers than the Confederate side.
A big likely contributing factor here would be the likelihood that the trend of slavery concentrating in the Deep South would continue. The Deep South crops of cotton, sugar, and indigo (especially cotton) are much more lucrative uses of slaves than the Border South crops of wheat and corn, or even tobacco, so practically from the invention of the cotton gin, supply and demand was leading the slave population to be "sold down the river" to the Deep South. The Atlantic slave trade will probably stay dead, so expanding cotton growing will continue to drain the slave populations of the Border South.
Add to this the OTL pre-1860 trend of slave escapes being much easier from Border South states, where it was a much shorter distance to free states where covert aid could be obtained and then to Canada where you're legally free once you cross the border. The effect would be magnified in an independent Confederacy, where the USA would almost certainly refuse to recapture and ship back escaped Confederate slaves and would be fairly likely to abolish slavery altogether within its own borders. As a result, 1) escapes northwards produced a further drain on the slave population of border states, and 2) slaveowners in Border South states often felt the need to offer inducements to their slaves to not run away, often in the form of a promise to manumit them after 5-10 years of good service. Both these effects were significant pre-1860 IOTL (especially in Kentucky and Missouri), and would likely accelerate once the Confederacy broke away.
Even by 1860, Kentucky, Missouri, western Virginia, and eastern Tennessee had low enough slave and slaveowner populations that they were only marginally connected to the Slave State bloc. As they continue to see their slave populations decline (sold southwards, escaped northwards, or manumitted as a reward for not running away), they'll have less and less in common (culturally, politically, and institutionally) with the Deep South that would dominate Confederate politics, and more and more in common with the free Northern states.
As each state breaks away, the remaining Confederacy will become weaker, and a new Free State/Union frontier will be exposed, pushing the row of "border states" southwards.