1) You'd have to specify whether this occurs after a long civil war or a short one. Very big differences for the starting state of the CSA and the attitudes of each. Also, does the CSA get Kentucky, Missouri, or the Indian Territory?
2) If you have a fairly long war -- long enough to let the CSA government become rather more centralized than some would like -- then you might have something. The issue I think is to make sure that the first secession is not Texas'. If Texas is the first to go, than I think the rump CSA is likely to consolidate. More likely I think is Georgia: let them secede over dispute with Richmond very soon after the war. Perhaps because Georgia refuses to send troops and aid in the event of a slave rebellion elsewhere. That rebellion is probably the key: let it happen in the Upper South, probably in Virginia. It festers and spawns movements in Tennesee and North Carolina (and KY if they're in the CSA). This leads Georgia and perhaps Alabama and Mississippi to secede from the CSA and form their own government. South Carolina withdraws, seeking to shelter itself from the growing "negro anarchy", but they prefer to go their own way rather than go with Georgia's group which by now might include Florida.
3) In response, Texas is probably quick to restore the Republic. The question is what Louisiana and Arkansas do.
4) Robert E. Lee (very sick) leads a cobbled together milita force in response to the slave rebellion. He demands the emancipation of the slaves in Virginia and affiliated states: given their freedom, the slave rebellions fade. Lee warns that this freedom must be real and advocates against controls on the free blacks; this leads Virignia and the remaining Confederate states of NC, TN, and KY to adopt a Jim Crow system of segregation. Blacks have their rights, as long as they stay separate from whites. A new Constitution is adopted along these lines; the Convention occurs on the grouds of Arlington, Lee's Home, before Lee dies in 1871,
5) The hard part, I think, is then to get the rump CSA, the Georgian States, South Carolina, and Texas to want to re-join the Union and to get the Union to want them to re-join.
- Texas: Tensions with Mexico are high, but the biggest problem is Indian raids. Cattle ranching requires access to markets and most of those are in the US. As refugees from the slave rebellions come westwards and the example of the CSA proves, Texas enacts manumission by 1870, prompted by the need to international markets. The Panic of 1873 crushes the Texan economy, leading to an Annexation movement by 1880.
- With Texas' re-annexation, Arkansas and Louisiana -- whose status and allegienace have long been disputed -- seek to re-join the Union. The USA develops an active policy under James G. Blaine to offer carrots to "our Southern Sisters" in order to re-build the Union.
- The rump CSA (VA, KY, NC, and TN), devastated by the Boll Weevil, sees the rise of Grange politicians. These "Populists" are very different from Planter Democrats and Whigs, advocating for poor southern whites. The Panic of 1887 and the example of Texas seals the deal: the Old Dominion leads her kin back into the Union. She's helped along by the publication of The Last Testatment of Robert E. Lee: these were papers, written by Lee in the final months of his life, in which he regrets the War of Secession and the disunionism in the South. Withheld by his family, they are discovered in 1878 and published in 1880, an instant best-seller.
- The Association of Gulf States (GA, MS, AL, FL) is rife with problems. By 1885, it fragments as MS and AL fear they are increasingly the pawns of GA. They have kept their slaves -- or sought too. Rather than revolt, however, many of the Deep Southern slaves have escaped. Ironically, most sought refuge in the rump CSA. The CSA refused to return the runaways, its new Constitution and new philosophy now adamantly against any action that might threaten "the concordia of the races." (i.e. get the Blacks made enough they might take actions against ruling whites; Blacks aren't as organized as Whites fear them to be, but the fear is all that's necessary). The Panic of 1893 is the last straw: Populist politics and manumission finally win approval. Offers of economic aid from Washington overcome resentment and by 1896, only South Carolina remains outside the Union.
- South Carolina in 1896 still clings to slavery and grows increasingly harsh in its slave code. In 1897 a Slave Rebellion breaks out. A combination of fear and panic ensue: the only choice to ensure law and order -- ask for annexation.
The hard part of all of this is how the US begins to assimilate the re-annexed States. The proliferation of Slave Rebellions is perhaps a bit much, but I'm thinking this is the result of the Civil War having lasted longer than OTL. The history of Civil Rights is probably quite interesting: segregation is built on ensuring domestic peace in the New South, so it may be less oppressive than OTL and political rights may flow naturally. On the other hand, the fear of revolt may preclude such inclusionary actions.