Ok, took pictures this morning. The mystery plant has smooth leaves where the horseherb leaf is rough.

These first two pictures are horseherb:

The next two are the mystery plant:

And this is a clipping of each side by side:

It reminds me of thyme, is it fragrant?

Yes, it does look a lot like thyme also. I has a bit of a bitter smell, not like thyme.

I added another photo.

Maybe. I think the blooms are different though. The one in my yard seems to have a bloom on a separate stem.

Believe the nodeweed pedicels lengthen as the flower matures like this photo sort of shows; or do you mean the flowers on your plants are on a separate stem from the ground?

I think you are right then, that's probably what this is. Here is a photo of the flower parts. Does that look right for nodeweed? And if so, is it common in TX? Or how did I get so much of it?

Odd, I was just searching Synedrella on the usda database, and it came up with 4 results, Calyptocarpus vialis (horseherb), with subset Synedrella vialis, Synedrella Gaertn, and Synedrella nodiflora. Out of curiosity, I clicked on Calyptocarpus vialis, and the map shows it as non-native in the U.S. but wildflower.org, npsot.org, and an abstract written by a man in Ft. Worth that discusses that very question.

I read there was a horseherb look-alike from the Americas called Cinderella weed in some Australian article/site while looking for info about acacias. The Cinderella/Synedrella naming was apparently cute enough to stick - but that is almost the limit of my knowledge. There is good support that they are different plants, but today's image search gives the impression that there's a lot of Calyptocarpus vialis around the world that has been mistagged as Synedrella nudiflora. Tried to find clear distinctions, but the few were ad hoc summaries of summaries. This one from a thread on DG is probably as good (poor?) as any:

• Synedrella: Plants annual, erect or ascending, not rooting at nodes;
  heads aggregated; disc florets 4-12; cypsellae of ray florets with
  lacerate winged margins.
• Calyptocarpus: Perennial, prostrate or decumbent, rooting at nodes;
  heads solitary at each node; disc florets 10-20; cypsellae of ray
  florets not winged

Am thinking your imitator may actually be something more akin to the thyme suggestion.

I had accepted what I read about horseherb being native to south and central Texas (NPIN, etc), so never bothered to look at its non-native status on the USDA distribution map... very interesting they don't appear to have it tagged as native to the US anywhere.

Well, whatever it is, I don't want it. When it gets bigger, it is lanky and leaves are sparse. It's odd that I've never had it before and now the whole side of the bed is full of it.

With all that flooding we had in May, there's really no telling what we have in our yards now. I was praying that the folks uphill weren't big herbicide/pesticide users. We had a river coming down the hill a couple of times.

That's true Pam, and I'm at the bottom of the hill.

As am I !!!!