I think the state most likely to get involved in a Confederate secession crisis is Texas. Texas was an independent country for nearly a decade, and Texans hadn't forgotten. It's less about looking back on the (questionable) successes of Republican Texas than the inspiration it provides. Texas is also large, very likely the westernmost Confederate state, far from the centers of power.
There are a couple of good friction points. The Comanches ran rampant in Texas during the Civil War, and this was a long running issue. OTL during the next decade, the Feds prevented the Texans from taking action and took little effective action themselves. I don't know that the Confederacy will try to stop Texas from attacking the Comanches, but I don't think it's in a great position to assist. It certainly isn't going to be stationing soldiers in the Union forts that were abandoned when the war started. The biggest effect of the Comanche wars was psychological - they were killing hundreds a year, but thousands fled. I think the Texans will take them out in the end, maybe even sooner, but it may spark the same kind of resentment that OTL Texas felt against the Union. And the fact that Texas accomplished it alone will reinforce ideas of independence.
There's also Mexico. A Confederate victory will probably skew Mexican history from there on in, but it's a fair estimate to say Mexico will remain troubled in some way for the foreseeable future. I can easily see cross-border raids into Texas that the Texans can't adequately respond to since their own forces can't strike back over the border. And if they do, the distant Confederal government is more likely to try and stop Texas than start a war over some stolen horses and burned-out border towns. (One could also envision a similar dynamic with the Comanches - conceivably, the US could purposefully let them use US territory to regroup beyond Texan reach. OTL the distant Easterners, Northern and Southern, felt a lot more sympathy for the Comanches - the natives of their own areas having been suppressed, exiled, or exterminated long ago).
And then there's oil. After all of the above, as Texas is left alone to rot, the Confederacy did nothing. But when the oil comes, they're going to want a piece of it. Or if nothing else, some kind of regulatory control - the OTL early unregulated Texas oil industry was an economic disaster that required Federal intervention and birthed an active culture of oil smugglers. That will be even easier when Texas shares an enormous border with two other nations, as well as its huge coastline. Ignoring inconvenient Confederate laws becomes ever more common.
To apply these in just one scenario, I can imagine a Texas that had to fight a long, basically independent guerrilla war against American-supported Comanches. The Rangers were turned into a more permanent, formal institution after Confederate independence, its outlines laid down by war veterans and sharpened by the extreme violence of anti-Comanche actions. Similar to OTL, early in the 20th century there's some kind of Mexican Revolution. Semi-ideological bandit freedom fighters in the mold of Pancho Villa are raiding across the border. As in OTL, Texas is in a state of heightened paranoia. Again, the weak and distant Confederal government does nothing. Texan-led retaliation raids over the border (which hit regular Mexicans more than actual bandits) and state-sponsored suppression and violence against Tejanos feed into the cycle. Maybe a José T. Canales analogue is raising heck over it in Richmond, although I don't know that the Confederal government would care. At the same time as this is going on, the central government is trying to crack down on oil smuggling and control prices and extract taxes. Texans feel they're paying out the nose and getting nothing in return, as usual (or so they claim). Then a European war raises its ugly head, and a paranoid, heavily armed Texas in the midst of severe internal troubles is told it needs to send all its men thousands of miles away to fight some foreigner's war. Under the leadership of a popular and charismatic governor, they decline. One thing leads to another, and that leads to the Second Texas Revolution.
