Adding where you are (nearest large city & state) and your USDA or Sunset growing zones is important to finding plants that will flourish.

How tall do you need something in that area? Would a woody vine such as bougainvillea on a trellis that reaches above the fence work?

Carrotwood is a prolific seed producer. Bougainvillea is typically (though not always) spiny and drops papery bracts continuously. Those are traits to avoid around a pool if possible. Maybe another palm similar to the one you have? I can't give a positive ID but it looks like Mediterranean Fan Palm (Chamaerops humilus). They grow multiple trunks, but you can remove those you don't want. They don't drop a lot of litter.

How much space do you have to work with?

Yeah, the look of bougainvilleas would be great, but they are very messy with "flowers" falling everywhere all the time. The only time when mine is not pumping out flowers is after it freezes. Then there are leaves falling.

Question for Kris, why don't you just get another like palm? It doesn't look like you have much space. If you are just trying to screen out the white structure beyond, it might be enough.

Giant bird of paradise, (Strelitzia nicolai) has the advantage of not dropping tons of small debris. It has the disadvantage of being a suckering plant, meaning it sends out new trunks from the base. They are lovely if you can keep them under control. I would have concerns about the trunks leaning against your wooden fence. I have one I have to cut back regularly. One trunk was leaning against my deck heavily enough to begin to push part some of the framing. Caution advised. Minimum planting area maybe 6' x 6'.

Nice palms you can grow that won't be super messy: 1) the kind you already have -- Mediterranean Fan Palm. It grows slowly, easy for you to anticipate direction of growth. 2) Triangle palm (Dypsis decaryi) likes sun, has self-cleaning leaves (they dry up and drop freely to the ground on their own) but most people will trim them before they fall. 3) Bismarckia nobilis, a silver-blue-green palm. Slow at first, picks up speed as it gets the roots established, eventually has a large crown of pleated fan leaves. Some trimming needed after fans fall down along the trunk. 4) Chinese Fan Palm (Livistona chinensis) prettiest in shade, large green fans with relaxed tips, has to be trimmed when fans dry. Stems have toothed edges, but not dangerously so. 5) If your climate doesn't freeze, Kentia palm (Howea forsteriana) has a very fine tropical look with green slender trunks and graceful fronds. Likes to start in some shade and grow into the sun. Big box stores often sell large pots with three planted in a group which will make a nice screen. Moderate speed of growth, need regular water, won't do well in extreme desert conditions. 6) King palm (Archontophoenix cunninghamiana) green trunks turn to grey with age, nice full green feathery crown of leaves,will show damage at temps below 28F -- know your climate before planting. Needs regular water to look best, but can tolerate some short periods of dryness. Self cleaning leaves; you may want to cut fruit before it matures and drops (small round fruits).