The area in question

I think we need a bunch more info. I find it difficult to tell where your property ends and how much space is available. Does your property go to where the fences change from chain link to wood? Do you want to design the entire fence line at once rather than doing the corner and later the zinnia area? Are you open to reducing the grass and making the bed deeper? Is there enough space for a smaller tree that will reach out into the yard a ways, and is there room (and would the neighbors be OK) to have some tree branches over the fence? Are you interested in year round blocking of the neighbor's garage or just in summer? Can we see more photos that give a better feel for the space as a whole or even something like a view of the property from Google maps or a plot plan? Right now I don't know how far the bed reaches along the garage and what happens out of sight along the left side of the photo. Do you have a preference for everything being the same like the zinnias, or would you like to mix things up a bit and have a mix of plants?

Hollyhocks are a good contender, thanks for that suggestion, littlebug!

Here's some more information about the area:

My property corner ends where the fence changes. I don't think the neighbors would have a problem with a small tree, but there are overhead utilities that run above the chain link, so it would have to remain small (15 feet tops.

To the left of the zinnias is an old honeysuckle and a hawthorn tree, then beyond that, raspberries are on their way to filling in the rest of the fence line.

Hopefully the photo below helps give a better picture. I have some experimental bean trellises going on back there and a flower bed in transition in the foreground. (Please don't inspect those to closely. :) )

The zinnia bed is 3' deep by 13' wide. It's a pretty tight fit against the hydrangea, but I wouldn't mind it growing deeper toward the left (like, extending out around the honeysuckle).

I think the new plantings should be a different height than the hydrangea so as not to compete visually, but the low zinnia planting as seen here reveals too much of the garage behind, so I'm thinking high is better. I'm not adamant about coverage in the winter.

Why not something to cover the entire length of chain link fence - climbing hydrangea? - and give a handsome background for everything in front? It can be trained to grow taller where you want to hide the garage.

Thanks for the additional photos. Are you willing/able to control Annabelle's suckering tendencies over in that corner? If not, whatever you plant will be competing with the vigorous stems of Annabelle. I had stems of Annabelle come up on the far side of some spireas; in my garden she really wants to take over the world, and only a larger, dense plant discourages her domination tendencies. I have a large rhodie that keeps her in check on one side quite effectively. I think I would plant something perennial or woody and somewhat vigorous as well as going out semiannually to pull any sprouts from Annabelle that reach toward that corner.

I think the goal here should be to create a bed that doesn't totally cover the garage, but makes it fade into the background by giving something of greater interest in the foreground. To totally cover the garage, you will create a wall of greenery, but having plants that partly obscure, but mostly just make the view focus on the bed itself will be enough to meet your goals.

I would put a large shrub or small tree several feet out from the property corner. Look at some of the small disease-resistant crabapples such as 'Pink Princess', 'Firebird', or Malus sargentii and its cultivar, 'Tina'. All stay under 10'-12' and have excellent resistance to all of the main diseases that effect crabs. Or one of the large viburnums like one of the many forms of V. plicatum, a large shrub that could be pruned up to tree form if so desired, would have blooms before Annabelle and red fall color to lengthen the season of interest.

If you'd prefer a perennial, look at some of the larger Coreopsis such as Coreopsis tripteris or one of the tall perennial sunflowers or Veronicastrum/culver's root or Joe-Pye-Weed, some of which will need dead-heading if you don't want a lot of seeding around. There are probably better suggestions out there for tall vigorous perennials; you could ask on the perennial forum.

For the zinnia bed, I'd use a set of your handsome trellis/obelisks, covered with a widely spaced wire grid and grow perennial vines on them. Look at the many viticella group hybrid clematis which need pruning to the ground once in winter or early spring, but otherwise are fairly easy IME. Make the obelisks 8'-10' tall, and choose some of the larger, longer blooming varieties; the clematis forum can make specific suggestions as to favorites. Alternatively, look at some of the non-invasive kinds of honeysuckle vines such as Lonicera sempervirens (I really like Major Wheeler), L. periclymenum, L. heckrottii (goldflame), or Lonicera tragophylla x brownii 'Mandarin'. At the feet of the vines and shrubs or trees, plant a groundcover that will bloom when the other plants don't such as Veronica 'Georgia Blue' which blooms in mid-late spring and then has tidy foliage with a few scattered blooms the rest of the season. It's semi-evergreen here, so nice well into the fall, and will spread and bloom in sun or shade.

I don't think I would want climbing hydrangea on a chain link since over time if it grows through the chain link it can damage it, and its clinging mechanism works better on a solid surface than wire. I like situating climbing hydrangea where it's a plant-and-let-it-grow, no fuss plant. A masonry wall or large tree trunk works better IMO.

I also don't think I would limb up your current large shrub (is that the hawthorn?) as severely as in Yardvaark's suggestion since that defeats the goal of camouflaging the garage. I would keep branches below the level of the fence top if there are any that low, trim back the front if it wants to lean out over the lawn (or change the bed line to accommodate it) and even out the growth somewhat so it looks more balanced. If the honeysuckle is to the right of the larger shrub and it wasn't intentionally planted, I'd probably remove it. Many honeysuckle shrubs are invasive (you can get an ID on yours on the Name that Plant forum) as the birds eat the berries and then spread seeds wherever they fly to, and design-wise that area looks fairly off-balance.

Yardvaark: You've inspired me to pull out MS Paint and play with your suggestions. Thank you!

NHBabs: Wow, so much great information! I'm glad you mentioned the Annabelle sucker issue--I'll take that into account. I really appreciate all you're suggestions and guidance on this!