Here's three:

Bignonia capreolata (Crossvine) is a native, evergreen or semi-evergreen (depending on the cultivar and conditions) vine with beautiful fragrant flowers. 'Atrosanguinea', 'Jekyll', 'Shalimar Red', and 'Tangerine Beauty' are some popular cultivars. BTW, stay away from TyTy Nursery. They often advertise crossvine, but are not a reputable nursery. Beaver Creek Nursery, here in Knoxville, has three of the four cultivars I just mentioned.

Clematis armandii (Armand's Clematis) is a reliably evergreen clematis. It's not a native, but has no invasive tendencies.

Gelsemium sempervirens (Carolina Yellow Jessamine) is another native, evergreen vine with pretty yellow flowers. 'Butterscotch' is a reblooming cultivar you might consider.