New data shows that fewer migrant children arrived alone at the southern border last month.

The number of migrant children and teenagers arriving alone at the United States border with Mexico decreased last month compared to a month earlier, according to newly released Customs and Border Protection data.

There was a slight increase in the number of border crossings, encounters and apprehensions overall during the same time period, a sign that the record surge of migrants trying to get into the country this spring could be starting to stabilize.

But the problem is far from over for the Biden administration, which is currently trying to safely place more than 16,000 migrant children in government custody with family members living in the United States. The administration on Monday threatened to sue the state of Texas if Gov. Greg Abbott, a Republican, follows through with his threat to shut down more than 50 shelters in the state where thousands of migrant children have been living.

Mr. Abbott’s action, which was part of a disaster order issued at the end of last month, was seen by many as a deliberate swipe at the Biden administration’s more compassionate posture on immigration compared to the restrictive measures of the Trump administration.

It is typical for the number of migrants traveling to the United States through the southern border to increase during spring months, but this year the turnout has been much higher, with a nearly 50 percent increase in border crossings, encounters and apprehensions in March, April and May compared to a similar surge over the same period in 2019.

Republicans have seized on the surge along the southern border, calling it a crisis — a term the Biden administration has avoided.

Most of the adult migrants who have been arriving at the southern border this year have been barred from entering the country because of a public health rule put in place during the Trump administration, which is responsible for more than 463,000 expulsions on the southern border between January and May of this year.

While the last administration also barred children for public health reasons, the Biden administration has been allowing migrant children to enter the country and stay in shelters overseen by the Department of Health and Human Services until they can be placed with a family member or other sponsor. Since the beginning of the year, more than 65,000 migrant children and teenagers arrived alone on the southern border, with record numbers arriving during the spring months. Nearly 2,900 fewer migrant children arrived alone at the southern border in May compared to a month earlier.

Because of a shortage of shelter space at the federal government’s network of state-licensed facilities earlier this year, migrant children were forced to stay in overcrowded holding cells along the southern border long past the legal limit. Earlier this year, the Biden administration moved to set up about a dozen emergency shelters where the children could stay in Health and Human Services custody until they are placed with a family member or sponsor inside the United States.

Recently, migrant children and teenagers have been staying in H.H.S. custody for an average of 37 days, according to government statistics. Children’s advocates have said ideally a child would not have to stay more than 20 days in a government shelter.