kjmcd,
Try an annual that is delicate looking when it grows, like cypress vine.
You can buy the seeds at any big box store.
I think they bloom by the end of July, little cute red flwoers, and the vine itself is lacy, delicte, pretty actually, different looking, not overpowering.
Look up online at pics.

What about carolina jasmine? Pretty yellow flowers and it is evergreen.

Oops, BTW that is jessamine not jasmine.
Another one to consider is Clematis Armandii, evergreen and fragrant white flowers in the early spring.

Will Carolina Jessamine climb on a metal chain link fence in full NC summer sun? Will the metal get too hot and burn the vine. What about Aster vine? I'm okay with putting a wooden trellis up in front of the metal fence to avoid killing my plants if need be.