It might be a corky-stemmed passion vine, passiflora suberosa. It's the host plant for gulf frittillary and zebra long-wing butterflies!

That's what it looks like to me as well.

I find little bits of this in my yard, say, 10-inch lengths. I'd like it to flourish. Any advice?

Just give it a trellis or something to climb up. This vine pops up around my property but the only ones that get sizable are the ones with plenty of support to climb up. The biggest one I have is only 2 years old but it already densely covers a very large trellis. It looks just like English ivy now. I don't do anything to it

yes, it's a Passiflora suberosa Sweet. it will overtake the small plant it's climbing on. you may want to consider moving it. it will just keep climbing up the plant to the top and then fall over and spread out. i have one vine that gets to be fifteen foot by ten foot ... covering everything around it and the plant it's growing on ... it happens to be arrow bamboo so i'm not concerned with it, but you probably don't want to lose your plant. ooh the butterflies will be so happy when they find it. :)