In New York, the Mayor’s Office of Immigrant Affairs was partnering with CUNY Citizenship Now! to offer two free clinics on Thursday and next Tuesday, while Make the Road New York, another immigrant advocacy group in the city, had filled all its slots for three Saturday events.

In Phoenix, a community organizing and advocacy group, Puente Arizona, had already helped prepare six applications on Monday, even though it was a federal holiday. On Tuesday the applicants only needed to put them in the mail — with a tracking number to avoid any problems with the United States Postal Service that plagued the renewal program in the fall, when more than 1,900 applicants were rejected for being late because of mail delays.

In some cities, though, the DACA push in response to immigration authorities reopening renewal applications was slower than others. Geoffrey Hoffman, the director of the immigration clinic at the University of Houston Law Center, said on Tuesday that the clinic had received only a handful of inquiries from DACA recipients.

“It’s been very sparse,” Mr. Hoffman said on Tuesday. “I would say there’s still a fair amount of uncertainty with regard to whether these applications will be accepted,” he added.

But another legal services group in Houston said it had received 50 to 100 emails, calls and walk-in consultations since the federal judge’s injunction last week.

On Sept. 5, when the Trump administration announced that it was phasing out the DACA program, which grants the deportation protection and work permits to young people who were brought to the United States illegally by their parents as children, it allowed only those recipients whose benefits expired between Sept. 5, 2017 and March 5, 2018 to renew for a final two years.

According to the immigration agency, 132,000 out of the 154,000 people eligible to renew applied in the fall. But this time, anyone whose permit had expired since Sept. 5, 2016 or was canceled at any time could apply.