I have a
Clematis 'Praecox' that really wants a 3'x30' fence--it climbs a little but really is a scrambling 'ground cover'--long bloom period thru summer when many other clematis are resting in my climate
It might be a little intrusive across the fence
What is the fence made of?
There are many more mild mannered clematis and I'd think you could try several
Clematis 'Arabella' I highly recommend
Star River is awesome too
The tall climbing varieties would work fine if you keep up with attaching them to the fence to go sideways, but they won't necessarily do that all of their own accord...