I would certainly consider Madison Star Jasmine.

Here is a link that might be useful: Star Jasmine

I was wondering what was keeping Melvalena's jasmine so upright. The close-up shows it has been trained on chicken wire. It would need some help scaling a fence. You could get some of that lattice material (wood, not plastic) at Home Depot, and secure it to the fence on the 4x4 uprights with small wood blocks. You could then attach the runners to the lattice with ties, to get it started upwards. Left to its own devices, it tends to grow willy-nilly. No plant more elegant, and fragrant, in early summer.

Trumpet vine. Thick and beautiful

Thank you! Yes, I very much love flowering vines. I have another crossvine I'm baby-ing in a container until I can figure out where to put it. I know it would certainly grow nice and thick and provide more privacy. I love love love jasmine! I have a star jasmine in a large planter as well. I definitely like the "wall o' vines" idea! Would work for my needs and would be gorgeous once it has filled out.

A Lady Banks rose (rosa banksiae) might actually work on the wider side if you don't mind all the trimming and training effort, especially during it's spring growth spurt. At least it doesn't have thorns. We had one do well in a similar spot for several years before it was removed for construction. It was semi-evergreen for us in 8a, though may not retain many leaves if you are much colder.

Here is a link that might be useful: Lady Banks Rose (Weston Gardens)

"You can't hiiiide
Yore pryin' eyeeees"

The lack of privacy here would trouble me. I'd want something of uniform density, and rapid growth, against that fence. Japanese boxwood, maybe? But it's so dated--so "70's." It would excite no HGTV-type "wow's." Simple, humble, unassuming. It would get the job done, though. You can always extend the bed, and do a little creative "layering."

HA!!! Thank you whitecap, I think I also would prefer something fairly uniform. The bare fence does bother me quite a bit. When I'm sitting on my patio, I feel like they can hear and see everything. They likely can. Their fence backed up against ours is just wire hurricane fencing. I also like the idea of planting roses there but that would require more maintenance than I'd be up for. Plus, the neighbor's trees, while pretty, litter all over our yard and I end up pulling tons of sprouts. I don't even know what kind of trees those are. Hm, maybe I could plant something like cleyera or wax myrtle along the fence? Cherry laurel?

I killed a cleyera once, and didn't even have to work at it. Just couldn't keep enough water on it. I've had no experience with the other candidates. Hollies would look good, but they're so slooow. Be sure to leave enough room between the shrubs and the fence to allow replacement of pickets, when necessary.

I was going to say Cherry Laurel! They grow pretty fast.

As a boy, I used to enjoy devouring the small, black, bittersweet "cherries" produced by a small tree in a thicket near my East Texas home. I suppose it must have been a cherry laurel. No one told me the fruit is poisonous--foliage too. Related to cyanide, as I understand it. They seem to be a bit pricey. I don't recall seeing any locally priced at less than $50, always in larger containers.

As for waxmyrtle, well, it would require a lot of elbow room or a lot of pruning.

That is a challenging space. I like the vines on lattice idea, but I'd put them in a bed so that you wouldn't have to weed eat around them. Plus then you could add some low shrubs in front of the vines and some annuals in front of the shrubs for color. Or you could do vines and big shrubs for extra privacy.

lynnemarie, I like that suggestion as well. Do you think a raised bed would be a good idea? I've given thought to having a raised bed made out of flagstone or brick installed along the fence and filling it with privacy shrubs. Adding vines and annuals/perennials would be lovely! Definitely better than this vast, blank expanse of fence.

Cross vine. Well behaved and doesn't send up runners all over the yard like trunpet vine (the WORST!).

Consider adding a paver-or-patio-stone-and-gravel patio outside the doorway -- which could create a nice area in front of the garden shed.

Then add stacking-stone raised beds by the fence -- and then add some taller lattice to the fence itself -- and then add plants .....

Here is a link that might be useful: Pinterest idea for a walkway -- but would work for a patio too!

We have a large corner lot with picket fencing all around as well. I have dug garden beds about three feet out form the fence and amended the clay soil. I have planted a few small evergreens, some boxwood, tons of clematis, flowering shrubs etc. There are so many options to go with. I like mixing a little everygreen for more privacy, with vines and flowering shrubs. I think a raised bed would be ideal for you, and you could amend your clay soil, by just adding raised bed mix, compost, good top soil, etc, after you remove the grass. Two to three feet maximum from the fence or you will have to get in the bed for upkeep.