OK, any idea what it is, then?

No, sorry. Texas flora is a long way out of my area.

Ah, actually your leaf observation just triggered it in me...I believe this is Wafer Ash (Ptelea trifoliata)!

And I did see another mature one in the area...so it makes sense!

Wafer ash has alternate leaves. Can't say for sure, but pic seems like it shows opposite leaves like ash.

The twigs look alternate to me. But the problem is that the pictures just aren't good enough to tell with any confidence.

Yes, the twigs look alternate to me, too.

And sorry, thin stems against a busy, non-contrasting background are always like trying to photograph Bigfoot... :D

If you could carry a sheet of white card or paper with you, hold it behind the twig and take a close up, we could see details of buds and leaf arrangement. Also try to avoid birds eye views.