Oh, there was plenty of that, not to worry. That's why we have a state of West Virginia, for starters. There were similar Unionist uprisings all over Appalachia, in Eastern Tennessee, Texas (interesting stories there about German immigrants from the failed 1848 uprisings who were about the staunchest abolitionists you could find anywhere, and the lengths the Confederates had to go to to keep them down), parts of Louisiana, too, I think. And those were just the biggest pockets of resistance, I've definitely read more than once around here that between the Unionists and the slaves, secessionists were a minority in the CSA alone.
And apart from people who wanted to return to the Union, the CSA had other internal problems, since they were actually quite terrible on the "states' rights" thing, and South Carolina wound up threatening to secede from the Confederacy, despite secession being explicitly forbidden in the Confederate constitution. Yeah, it was a mess down there, for sure.