Quick privacy requires building. Add a lattice work trellis to the side of your deck to give immediate privacy. If the lattice is not sufficient, add vines to further obscure the view. The upright posts should be sunk into the soil, so 10' posts will give you about 8' of trellis height. Two or three cross pieces between the uprights will help to keep the uprights aligned. Treated wood is notorious for warping, so here is one LINK that talks about prevention. Trellis panels are 4'x8', so you should position your uprights accordingly. You can also have someone make wider panels for a custom size.

Cherry Laurels were a good choice. I'd go to a nursery and look into other upright shrubs that do well in your area to use on the side yard where your neighbors house is. Add a tree or two, but leave enough room so it doesn't grow over the fence line. You may be able to place the magnolia closer to your patio so it blocks the neighbors house.

Where do you require privacy? On the deck or in the yard? If it's on the deck, you might have some limited options.

Here are some options for your deck privacy.

An retractable awning can provide some privacy also.

That fence looks very dry and the overall appearance looks like a 'shanty town.'. Might be time to get some stain on the fence??

Any update on your progress? I just went to the nursery and looked at Cherry Laurel, needlepoint, and Taylor Juniper

I went and looking for yew plum pine too. But that one was quickly remove from list. It looks very beautiful online, but doesn’t do well under direct sun, especially the Texas heat. They were struggling at the nursery.

If looking for a tree to provide privacy, Thuja “Green Giant” is a fast growing arborvitae that is suited for zone 8. Grows about 3’/year so start with the largest size your budget allows for quicker privacy. We have these in our yard and love them since they are tolerant of a variety of conditions and not bothered by many pests or diseases.

IMO trees are not great for privacy and that fence is a total mees so why not just doa new higher fince with allice on the top si 6" in from the old fence which I presume belongs to the house next door since yours is anew build, The if later you still want more privacy grow a vine that will cover the lattice . Go the max height youa re allowed and I seeno windows looking to your yard so the fence should be all you need . IMO privacy outside is all in your head I do not think neighbors spane their day trying to see what you are doing and the one house far awy would need binoculars to even begin to see into your yard. Any tree you plant there will overtake the space for sure.I think the back fence is so close to the neighbor I would add the layttice to the top of that one too

For many people, myself included, a garden without privacy isn't a garden at all, I agree that a new fence, if affordable, would be great to block the ugly existing fence. Then some shrubs and small trees (the nursery can tell you what grows well in your area) to soften the look would be lovely. If a new fence is too expensive, just do the tree and shrub border. It will block out the fence and the neighbors in time.

Actually arborvitae are a great source for privacy and grow much taller than the average fence height....and it's cheaper. You just have to have some patience to let them grow and fill in the space. While your fence has weathered it doesn't appear to be in bad condition. Cleaning it will take some elbow grease but it's worth doing. Much cheaper than replacing even if you hire someone to clean it for you. You don't have to line your entire lot line with these, just in areas where you need more privacy to protect neighbors fron viewing you. A variety of trees and shrubs keeps things interesting. You have that Texas sun beating down on your home, so be sure to think through the height of the trees and what will offer you sun protection at different times of the day so you can enjoy your nice big yard even in the summer months.

OOPS did not check the date.