When Gov. Andrew M. Cuomo called for medical workers around the country to come to New York last month and join the fight against the coronavirus, Bevin Strickland was ready to help.

Ms. Strickland, a former pediatric intensive care unit nurse in High Point, N.C., spent hours trying to submit her volunteer application online, and then emailed city and state representatives. She never heard back.

Frustrated, she reached out directly to Mount Sinai Queens hospital in New York City. A manager told her to use a private recruiting agency, which the hospital had used for years to bring in temporary staff.

Within two days, Ms. Strickland, 47, received her assignment. She started this week in the hospital’s emergency department, making about $3,800 a week for three 12-hour shifts instead of doing it for free, as she had initially wanted.