Are you talking about certain areas/counties in some of the seceding states (e.g AL, MS, TN) where people opposed secession and rebelled or otherwise refused to participate in the Confederacy? I haven't heard of a state seceding from the confederacy.

On people moving: As things are today, it's too expensive for many people to be able to relocate as a practical matter, but if things ever deteriorated to the degree we're talking about, the stakes change drastically. If there's a war, or some people are about to get trapped where they don't want to or are afraid to be, it's no longer "can I find a decent job and place to live there and afford moving expenses?" but becomes "I've got to get out of here any way I can even if I can't take anything with me and don't know how I'll get by once I'm there." Especially people who fear for their lives or safety in a country where they are hated or considered less than human by the majority.