I think
@Danita is correct in that you are closer to zone 7.
Without knowing what kind of sun exposure the fence line gets I am going to suggest Bottlebrush tree. This fast growing shrub has beautiful bloom, evergreen slender leaves and is drought tolerant once established. If you know someone who has one of these you can just cut branches, strip off the bottom leaves and some of the bark and stick them in the soil and water them in. Right now is the perfect time for doing this. The bloom attracts hummingbirds and butterflies.
Gardenia and Glossy Abelia are two more choices.
Another thing to consider is Leyland Cypress. These are fast growing trees and provide privacy, but they are evergreen as well.
All of the above suggestions from the other members of this site are excellent and do remember to plant away from the fence to allow room for maturity. There are species of Azalea's that bloom all year now. They are attractive and will get 10 ft tall eventually, but they are not going to reach 6 ft for a good amount of years.
Oh, I almost forgot to mention grasses. They grow fast, they are lovely, and there are many kinds to choose from. I am partial to Red Fountain Grass, Miscanthus, and Pink Muhley varieties. A grass garden is just perfect for molding a landscape and it is drought tolerant and you will only need to cut back the Red Fountain Grass in the winter close to the ground in order for the new green shoots to come forth. Hope this helps.