By what mechanism do you and ex have joint legal custody: court order or unincorporated separation agreement?
If by court order, ex can move for modification of existing order by claiming proposed relocation of child to TN constitutes a substantial change of circumstance adversely affecting child’s welfare. A judge will have to decide the “best interest” question and rule accordingly. So, you don’t really need ex’s permission, but you do need the court’s permission. If you don’t get it, ex could seek to have you held in contempt for non-compliance with physical custody (i.e. visitation) provisions of existing order. You and ex could, of course, try to settle your differences out of court via a consent order. I strongly advise you to at least attempt to come to agreement with ex.
If by unincorporated separation agreement, no, you don’t need his permission. You (or he) can take your daughter wherever you want. These agreements are largely unenforceable with regard to custody.
Your ex has no right or “standing” to make you stay or live anywhere. His rights extend only to custody of and visitation with your daughter. If a judge were to rule that child stays in NC, this ruling would not prevent you from moving to TN.