Does this happen around 1860, or are we looking at an earlier split? That matters, because it determines how divided the factions are, and thus also how likely the now-dominant South is to pass all sorts of legislation to re-make the remaining USA into its own image. That does, however, open another can of worms: for Northern secession to really become a thing, the South must already have gotten its way, unless we're talking about some earlier POD. Which probably means that more states than the ones you list will be seceding. The alternative is to stay in a Union that basically has the fugitive slave law enshrined as holy writ, which compels all states to respect slaves that are 'registred' in slave state as legal property that must be respected, and which has it legally determined that slavery cannot ever be abolished.
Given those kinds of circumstances, expect Ohio, Indiana, Illinois, Wisconscin, Minnesota, Iowa, Oregon and California to also secede-- and for the North to claim everything north of Kansas Territory and New Mexico Territory. (Although the North's claims on any non-state territory can probably be reduced to "everything north of the 42nd parallel".) You still get a huge "South"/Rump-USA, of course. But the idea that the states I listed are going to stay with a Southern-dominated USA under the sort of conditions that pretty much must exist for a Northern secession to even occur is simply not realistic. Furthermore, Britain -- in OTL unwilling to aid the South because slavery is disgusting -- would in this ATL see a very nice option to help the Americans divide themselves against each other, while helping the side that isn't tarnished by slavery. How very convenient.
No matter what its size ends up being, though, the rump-USA will presumably have gotten the following measures througth even before the secession (and this would in fact be what causes the secession):
-- Fugitive slave law in Constitutional.
-- All states must recognise the legality of slaves that are property in any slave state, and must not interfere with this if such property is transported through free states.
-- Slavery cannot be abolished except by any slave state, of its own accord.
In addition, post-secession, expect:
-- All the above to be written into the Constitution, along with a prohibition of future secession.
-- A decree, possibly also written into the constitution, determining that all future states shall be slave states.
-- A policy of government-backed filibustering, although that will probably fail because Britain's attitude will be: "Ha ha ha, none of that nonsense now, chaps! Wouldn't want the Royal Navy to have to land troops in Charleston after liberally bombarding the city, now would we?"
That's indeed not a recipe for a dictatorship, but one may expect the rump-USA to be fully committed to slavery, and its legal system set up in a way that makes abolishing it almost impossible. The increasing disgust of other powers will only lead to increased isolation. The best case scenario is actually that they stat being all jingoistic about their Caribbean "Golden Circle', and Britain just intervenes with military force, ends the existing regime, and abolishes slavery at bayonet-point. Good afternoon, and enjoy your further existence as a British client state.