DEL RIO, Texas—The Trump administration has curtailed a key component of its “zero tolerance” immigration policy, no longer charging first-time illegal border crossers with a crime along a busy stretch of West Texas.

Prosecutions of single migrant adults caught crossing the border for the first time in and around Del Rio, Texas, were suspended in February amid lack of jail space, said a U.S. Border Patrol official. The policy change hasn’t been previously reported.

Instead...