I love Little Gems. They are evergreen and bloom here for half the year (May through October). But, you can't walk under them as they sprawl and bud even at ground level. They are slower growing but may eventually be too big for the smaller space you are describing so may not what you really want. Might also consider Bay Laurel if your climate tends to be on the hot and arid side.

I went to a nursery and looked a few evergreen trees that don't get too large and the Photinia tree is what they seemed to recommend most.
It's supposed to grow about 15-20ft feet tall and have about the same width canopy.
They sell them in 15 gallon containers and they are already over 6ft tall. They are also supposed to grow quite fast so I can expect some usable shade in just a few years.

So, if at least half of the eventual 15 to 20ft canopy is directly over the patio and the other half is over the fence I would guess tat even a 8 ft overhang would still cast a big enough shadow so there are a couple of big shady spots to sit outside at each side of the patio and not rely solely on patio umbrellas for shade and it should also provide some privacy from people looking down from upstairs windows across the street...

Now, I wonder if the new shade from the one on the left side of the patio would cause growth problems for the Thuja Emerald Greens that are already planted near that location along the fence?

Just a few things to consider....... magnolias develop a lot of surface roots and evergreen forms like Little Gem drop foliage constantly. One IS able to locate them with a clear trunk, however.

Not sure where you are located but photinias in many parts of the country are extremely damaged by a leaf blight that results in almost complete defoliation, weakening and eventually killing the plant. Across much of the southeast and in the Pacific Northwest, photinias tend to be very limited in their availability because of this.

Zone 8 covers a lot of diverse territory, but I'd consider strawberry tree, Arbutus unedo, among possible choices.

I didn't notice any strawberry trees at the nursery, but don't they make a mess on the ground all year?
I'm on the west coast and the photonias are not hard to get. I went to two nurseries and both carry them. One was sold out.
I'm thinking about only getting one tree now because I'm concerned that the shade from a tree near the Thuja Emerald Greens would end up killing them because they are supposed to have full sun.

I keep reading negative things about the photinia, so now I'm thinking about the little gem magnolia again.
I would rather not have to deal with leaves falling, but maybe a small trickle of leaves falling year round will be better than a tree dumping all its leaves in winter and not having the privacy benefit of the tree again until the next spring.
Can't I keep it from growing surface roots if I do long watering sessions once or twice per week so the roots go deep?
They are a bit narrower than I had in mind for creating a couple of shady spots on the patio, but I assume they will cast a shadow larger that the actual spread of the tree, so it might be fine.
One of the reviews said it grows in compact spaces and are good for town-homes and to separate closely spaced homes. Kind of sounds like it should fit my requirements.

"Design Ideas:
Although it is labeled as dwarf, this very columnar evergreen will reach over 25 feet tall at maturity. A great plant to separate multistory city homes on narrow lots without crowding them. Makes a great evergreen screen to block unattractive views, increase privacy or absorb sound. With its pretty flowers, it makes a great accent tree for small space. Ideal for town houses and condominiums."

"I didn't notice any strawberry trees at the nursery, but don't they make a mess on the ground all year? "

Far less than does the evergreen magnolia :-))

Not sure where on the west coast you are located - if in the PNW, strawberry trees (Arbutus unedo) are by far your best choice. Photinia will contract leaf spot - only a matter of time - and the magnolia is easily damaged by any winter snowfall. If further south (CA?), you are more open to possibilities as the photinia is a lot less likely to develop leaf spot and you have no winter snow issues to speak of :-)

btw, the Indian hawthorn is not a good choice for the PNW either and will never achieve tree-like stature.

There is a cultivar of Live oak Quercus Virginiana called "hightower" or a similar name that is much narrower than the traditional Live oak of the south. A websearch should direct you to Southern pride tree farm. They have one named "Cathedral" also. They are located in the south, I don't know where, maybe Georgia or Florida. If you are interested you can search further. I found them on my search for different live oaks to try in zone 6. I have never bought from them, but remember they had narrow live oaks, and they came to mind when I read your post.