During the American Civil War, Texas was the last Confederate state to secede from the Union. What if, instead of joining the CSA, Texas decided to become an independent republic again? Perhaps the Texans wanted to keep slavery but didn't want to go to war with the Union, and sought to become a buffer state between the US and Mexico instead. Maybe they could reestablished separate relations with Mexico against the US. I doubt any European powers would be particularly interested in Texas, but maybe they'd get diplomatically involved with the French intervention in Mexico.
I don't know, just some ideas. How would this work?