The Classic, "Radical Reconstruction in full force would turn the South so reactionary it never considers itself American Again, be" line used by those such as Harry Turtledove himself could do the trick. The idea of the south constantly rebelling from the late 1860's on seems to always have a butterfly net around it too so it would be possible.
But then I don't ever consider that too plausible, and even if it is, it's far too clichéd for my tastes.
So basically I'd suggest that perhaps the option could be set a little later, in the 1870's. Have the "Reedemers" get a good kick or two in the face before eventually overthrowing the Republican governments of the South. A few more riots that get called "Battles" by old men but with them "adding defeats to the South" or somesuch. But lets say by the late 1880's we do in fact have the Solid South back.
Moving on to the 1910's, have America enter WWI but with someone besides Wilson or Teddy Roosevelt or anyone else who will use soft gloves on the South, someone who might be vilified by a stronger lost cause/klan/neo-South movement (More "vets" from the 1870s having given it a boost). Maybe the US gets in the war earlier, to have more men killed/wounded/mentally scarred. But for those who come home south of the Mason-Dixon line "Mr. Yankee X's War" was nothing but some brutal war that they had nothing as Southerners to do with. They're mad they fought a war for Yankee Industry and Yankee Bankers and all that.
Such a movement could grow, theres certainly room in the Solid South for a Third Party to rise up at the time, as they only have to face the Yellow Dog Democrats, though thats no easy task. Eventually have it develop into an Angry BQ sort of deal. Some way or another have the WWI Generation that are now the leaders of the movement not get too angry over WWII (Maybe their leadership is in the New Deal Coalition with FDR?) Iin the War somehow you have to keep alot of Southerners in Separate Units, perhaps recruitment in the South being more dominated by the National Guard then the regular Army.
After the Second World War another Generation of Southerners, come home with bitter feelings about the deaths in like with what their fathers felt. Maybe with Anger over "Yankee Miss-management" (Stick Southern Divisions in places like Italy and the South-west Pacific theater in 1944/45 and such feelings would be easy enough to develop) as well as Anger over the 1917 Generation for having "betrayed them". Southern Nationalism has a major and angry boost, come 1950 when the time comes for National Guard Units and Southern Boys to go fight another war in Asia, this time for an organization like the United Nations, they've had enough, and the WWII leadership Generation decides its time to take a stand and start a series of events that rapidly escalate into shooting.
Of course, all of that requires alot of butterfly nets to keep the Korea from even happening here. As it requires WWII to go just so and that requires alot more before.