Can a unmarried Texas mother move out of state while the father is in jail

It depends if the father's name is on the birth certificate and if the he's been ordered to pay support. In Wisconsin, you need to notify both the other parent and the court if you're moving out of state with the child; if the other parent doesn't approve, they can contest the matter and you'll have to wait and see what the court says.
Hope this helped!

Sorry to disagree but name on birth certificate and paying child support has nothing to do with it what so ever. It all depends on the child custody agreement that is in place, If the child custody agreement says you can't, you can not with out going back to court and getting it modified.

I only asked about the birth certificate because of parental rights. As in: if there's no proof that he's the father, would she even have to notify him that she was moving?

I brought up child support because if he is paying, then there would be an agreement, just like you said. That agreement would define what steps she would have to take if she wanted to move out of state with the child.

I probably should have been more thorough in my asking and explaining the first time *lol*

A child support agreement is separate from a custody/visitation agreement. The ONLY thing that matters here is whether there is a custody/visitation agreement in place. If, by moving, it makes it more difficult for the father to use his visitation then the mother would need to get court permission to move.

If there is no such agreement then the mother is free to move.