South Carolina is something of a special case, as it had an reactionary extremist streak to it for much of its early history. South Carolina provoked the first and second secession crises, the second of which of course led to the Civil War. It was also the last state to allow direct election of presidential electors, sometime in the 1860s after the Civil War.
South Carolina was the only state in the Union that had a Black majority. I don't remember if the ratio was 3:2 or 2:1, but it was high. If Santo Domingo and Nat Turner was on the minds of every White Southerner, then White South Carolinians had such visions in their nightmares. I don't know this for a fact, but it wouldn't surprise me at all if SC was the first state in the Antebellum South to make legally mandatory that all adult White males carry firearms at all times when traveling.
In some cases being a smaller minority group leads to less discrimination. The United States around the time of the Civil Rights movement might not be the best comparison, but I've heard that racism wasn't really the typical North and South divide that people often think of. Some people have said that they experienced worse racism in the North than in the South, mostly in areas with large minority populations, while it wasn't that bad in smaller towns where they might be only ethnic or religious minority.
If by the Civil Rights Movement you mean the 1950s and 60s, you didn't see public lynchings in the North using carefully posed photographs with everyone facing the camera so that there would be no doubt as to who participated in the murders. That was the South's way of saying that the Law had no place for the White Man exercising his Aryan Rights over the Black Man.
Murder is a state charge without a statute of limitations. Meaning that they were fully confident that if they lived to be a hundred they would never face justice for their crimes. They were right. No real civil rights laws in those days. About the only thing Eisenhower's anemic Civil Rights Law of 1956 accomplished (even only partly, with Ike still president and J. Edgar Hoover as head of the FBI) was to enable the US government to at least investigate such crimes.
Then you consider that the Confederacy is going to be focused on keeping down a large portion of its population that is enslaved or (later on) free but threatening to their continued dominance of the situation, and some of the racism might be relaxed. When you look at the history of colonies and even South Africa, the ruling elite (ironically in the Confederacy, African-Americans might be the majority) tends to expand membership to other groups, even some members of the majority group that is being oppressed. It might go from the WASP elite to the elite in general, then WASPs in general, and then perhaps out to ethnic and religious minorities in general.
The South African Whites are a bad example. The Afrikaners were overwhelmingly outnumbered by Black Africans, "Coloureds" of partial African descent, and Asian Indians. You also have to factor in among South African Whites a 35% British descended population who favored some kind of accommodation with the non-Whites of South Africa and enjoyed the same political voting rights as the Afrikaners while lacking their pathology about what they called "kaffirs".
Even in South Carolina you wouldn't have faced as bad a situation for SC Whites as the Afrikaners did.
I don't see too much reenslavement or enslavement occurring, because that could kick off something akin to the South African embargoes decades earlier. Enslavement of Catholics and white people certainly won't win the Confederacy favor among the European or Union public.
I just want to interject this: My own total rejection of the TL-191 Turtledovian argument of the CSA's longterm survival against a hostile USA, with the agrarian South repeatedly whipping (or at least surviving against) the industrialized and eventually even mechanized North using just "Southern Guts And Steel".
So talk of the Confederacy lasting well into the 20th century is interesting from an intellectual POV, but IMVHO such an idea is one step removed from Hitler conquering the whole of the British Empire.
Probably be more than "a family or two," though...
There was a lot of doth protesting too much among whites in the south at times...
Best,
Accusations of One Drop violations lasted far far beyond the ACW.
Babe Ruth was constantly heckled by unfriendly crowds for having "N----- Lips".
In one ball game, a player was accused of being "part-N-----" by a heckler in the stands. This, in a time that was extremely racist, the early 1900s, and made against a player,
Ty Cobb, considered to be supremely racist
even by the standards of the day! Cobb proceeded to jump into the stands, and started beating up the fan. When nearby fans pointed out to Cobb that the man he was pummeling mercilessly
had no hands, Cobb angrily shot back:
"That's right, and after what he called me if he had no feet I'd be kicking him too!". And he wore sharpened steel cleats.
Ty Cobb was easily the most hated man in baseball in his day, a true shit of a human being whose signature style was to slide into base with those steel cleats raised to try to spike (and possibly cripple) the player at the base or home plate. This was no secret to anyone, but being Ty Cobb in an era that preceded even Babe Ruth he got away with it.
When the baseball audience at the stadium realized what was happening, and what Cobb had been "accused of", they started
cheering him.
EDIT: Based on what DNA analysis is telling us today, with your average White American (of long term US ancestry) having some 15% Subsaharan ancestry, I'd say that the whole "One Drop Rule" would be headed for destruction eventually anyway. Even in an USM world where the CSA not only wins the ACW but is somehow allowed to survive to the present day. IMO most people who have claimed to have "a little Indian in them", even if they were not lying themselves, probably had an ancestor or two who "Jumped the Fence".