The uncultivated areas around Riviera and Baffin Bay Texas is dominated by mesquite trees. But a few of these are found here and there. Any thoughts?

An elderly Yaupon?

It reminds me of Ceanothus.

I am totally novice when it comes to this stuff. This tree also has small thorns that will easily pricked my finger when I pulled on its leaves. There was no discernable odor when the leaf was crushed. Here's another photo

Do you remember seeing them bloom?

No, I go to that site once a quarter. Last time I went was a few days ago that's when I noticed this and another tree (a different kind of mesquite?). I don't recall any of the trees flowering. Here is a more precise location in UTM: 14R 626580 m E, 3017264 m N (don't include m, N, or E when typing into GoogleEarth)

If it has thorns, perhaps it is Blackbrush.

I don't think its Blackthorn. What's popping up in Google images show thorns much longer than what this one has.

In the second photo I posted, there is a branch from the center of the picture pointing almost straight left. You can see the thorns pretty well.

What is the best information to share to aide the ID?

Perhaps one of the thorny Celtis species. Check out Celtis ehrenbergiana and see if that is similar.

Mayhaw ...or a species of (Crataegus aestivalis, Crataegus opaca, and Crataegus rufula)

KellyFW: Celtis ehrenbergiana looks like it might be the one
deepsouth: the Cratageus spp are a little different. Are Celtis and Cratageus is the same family or order?

Just a guess here, but the gray bark, leaf shape, leaf color and veining in your photos do look a lot like Texas Torchwood (Amyris texana), also called Chapotillo, endemic to South TX. You might Google/investigate that possibility to see if you think it's a match. I JUST bought a Torchwood from The National Butterfly Center, hoping it will go here in 8b Temple. They do have a website and might be able to ID your plant. My Torchwood arrived last week. They pruned it heavily for shipping, but this is what it looks like at present:
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They flower clusters of white flowers with yellow centers and produce a dark drupe fruit March-October intermittently that are edible to birds and humans. Have you ever seen these bloom? or show any fruit?

Peggy8b: I have not seen it blossom. But I also haven't noticed it before last week either. Most of the trees here are mesquite and from a distance mesquites and the tree in question look similar. I still think Celtis ehrenbergiana, suggested by KellyFW, is most likely

Also the tree in question does not appear to have compound leaves.

You're right, Porkpal.

Well this has turned out to be quite the discussion and fun to solve this with you all. I'll leave the thread open today for any other insight, if not, I'll mark it "solved, finished, or expired"