In 1680, the Pueblo Indians revolted against the Spaniards of New Mexico. The Spaniards were forced out of New Mexico into the area of El Paso, Texas. Many communities were formed; some named after communities in New Mexico. Socorro del Sur was one of them.
The people who lived in Socorro at the time were mostly Piro Indians. These people followed the Spanish south. A mission was built in Socorro del Sur called Nuestra Senora de la Limpia Concepcion de las Piros del Socorro [Our Lady of the Immaculate Conception of the Piros of Socorro.] Here is a link to a historical marker at the site, as well as photos of the church. I found the marker on the Historical Marker Database.
I also found this site about the Piro Indians.