Reservable USFS campgrounds in Texas are opening in early June, at least Recreation.gov is taking reservations for those dates.
Santa Fe NF in New Mexico is taking reservations starting June 15.
Ozark NF in Arkansas is taking reservations starting June 1.
Cedar Lake in the Ouachita NF in Oklahoma is taking reservations for the second week of June.
Apache Trout Campground in the Apache Sitgraves is apparently open now, but restrooms are closed, and there is no water available, no trash pickup. Carry out all your own trash. No tent camping allowed. Only RV's. Possible late June more services to be restored.
Deming NM paper reports thee additional recreation areas opened in the Gila NF
3 More Recreation Areas ReopenThis link has info from the Southwest Region of USFS for New Mexico
USFS Southwest Region 
