On a conference call Monday, lawyers who have fought the Trump administration argued that other refugees and travelers should also be allowed in because, like Mr. Dagoum, they often have ties to a nonprofit organization that has been helping them even before they land in the United States.

“Anyone who has an existing relationship with a nonprofit, frankly tens of thousands of refugees,” should be seen as having bona fide ties, said Becca Heller, director of the International Refugee Assistance Project.

Representatives of some resettlement agencies said they were awaiting guidance from the State Department. Heather Nauert, a State Department spokeswoman, said on Tuesday that the department would consult with the Justice Department on how to define “bona fide relationship,” a process she expected to take two more days. Meanwhile, anyone already approved for travel to the United States by July 6 would be allowed in, she said.

Mr. Trump, who hailed Monday’s decision, has said he issued the ban to give his administration time to review its vetting procedures, while opponents contend that the move discriminates against Muslims. The justices will consider the constitutionality of the ban in the fall, but in their order Monday, they signaled that foreigners without American ties were the ones least likely to gain any protection from the court.

Although the administration has not said how it would interpret “bona fide relationship,” it is conceivable that it will take a relatively narrow view of the phrase and decide that anyone without a family, university or employment connection can be barred.

That could lead to another round of lawsuits from opponents of the ban, the very situation that Justice Clarence Thomas warned of in a partial dissent in which he called the standard “unworkable.”