It was obvious the site was overwhelmed by the volume, said Mr. Connolly, who is worried about his wife and two young daughters. They were on his health insurance.

“As soon as I saw that, I realized that it was just going to be a nightmare,” he said.

He kept trying on Tuesday. It didn’t get better. “I’ve given up,” he said Wednesday.

Help is on the way. A stimulus bill that President Trump signed into law on Wednesday provides $1 billion in emergency grants to states for unemployment insurance, and fully half could be immediately used to strengthen staffing, technology and other administrative costs.

The need is urgent. In Colorado, the number of unemployment benefit claims went from 400 on March 9 to 3,900 on Monday and then 10,000 on Wednesday. In Connecticut, there were 2,500 claims last week, but more than 40,000 from Friday evening until Wednesday evening. The number of requests in Ohio jumped to 78,000 this week from 6,500 two weeks ago.

Maryland said that its website and call centers were experiencing “an unprecedented volume” of requests. The state’s division of unemployment insurance said it was extending its call center hours, working to increase its network’s bandwidth and asking claimants and employers to email.

New York’s online application system crashed as it tried to accommodate a spike in volume that resembled the surge after the 9/11 terrorist attacks. By noon on Tuesday, the state labor department had received more than 21,000 calls, compared with 2,000 a week earlier. Visits to its website more than doubled. The department is now extending its hours and staggering filing times alphabetically by last name.

Delta Vasquez, 22, spent several hours trying to slog through Oregon’s unemployment claims portal on Monday night after she was laid off from her hosting position at Bamboo Sushi in Portland. She and her co-workers got the news by email after the state moved to restrict bars and restaurants to takeout and delivery.