I guess I'm not using the right search terms, because I cannot find this thing on the internet. I'm calling it a vine, but it's actually a woody "shrub" that sends out long branches with thorns that grab onto nearby trees to use them to climb up. The first pic is a closeup I took yesterday. You can see it's berrying. In that pic, the vining plant is a honeysuckle, but on the branch you can see one of the thorns. In the second photo, is the same plant that used one of our pecan trees as a trellis. We cut it down several years ago but it grew back.
The flowers are small and inconspicuous but smell sweet. The berries are soft, not smooth, and the chickens like them.
Thank you, All! I followed your leads and believe it is Elaeagnus pungens, aka Thorny Olive or Silverthorn. It's sold as an ornamental, but listed as a non-native invasive that can be quite problematic. I understand why! Greg, I didn't see anything about edibility, but I think it's hopefully eradicated now. So many other useful things to grow!