DONNA, Texas—After tens of thousands of families streamed across America’s southern border late last year, a Catholic charity in Tucson, Ariz., opened a monastery to house asylum-seeking migrants whom federal agents released. The 66,000-square-foot space is now full. The recent discovery of an unused closet caused excitement.

An additional 113,797 people in families had crossed this year by the end of March, and border agents in El Paso, Texas, housed some under a bridge. Last month, Yuma, Ariz., declared an emergency after a converted strip-mall thrift store serving as a shelter hit capacity. Last week, a giant white tent opened on the outskirts of Donna, near the Rio Grande—the government’s latest desperate attempt to house the families.

In April, a single-month record of 58,000 parents and children crossing together illegally were arrested, federal data released Wednesday show.

A deepening humanitarian crisis on the border is straining towns, aid groups and federal agencies as thousands of families, when released, immediately need shelter.

The border has seen illegal-immigrant waves before. This one is different.