EAU CLAIRE, Wis. — As the coronavirus pandemic swelled around the 160-bed Mayo Clinic hospital, the day was dawning auspiciously. Two precious beds for new patients had opened overnight. At the morning “bed meeting,” prospects for a third looked promising.

Better yet, by midmorning, there were no patients in the Emergency Department. None. Even in normal times, a medium-size hospital like this can go many months without ever reaching zero.

Everyone knew better than to trust this good fortune. They were right.

From 9 a.m. to 10 a.m., seven patients arrived at the emergency room. Fourteen descended the next hour, then 10 more the hour after that.

About a third had signs of covid-19, the illness caused by the virus, most with trouble breathing. But there was also the man who had smashed his fingers with a hammer. The unresponsive woman who had to be resuscitated. An injured elbow. Neck pain. Acute depression.

By 12:05 p.m., Mayo had put itself on “bypass,” sending all ambulances to the two other hospitals in town, a last-resort move rarely employed. By late afternoon, the emergency room was stashing patients in four beds erected in the ambulance garage — the first time it had adopted that tactic — and holding others for hours as they waited for places in the overflowing hospital.

With more than 91,000 covid-19 patients in their beds, U.S. hospitals are in danger of buckling beneath the weight of the pandemic and the ongoing needs of other sick people. In small- and medium-size facilities like this hit hardest by the outbreak’s third wave, that means finding spots in ones and twos, rather than adding hundreds at a time as New York hospitals did when the coronavirus swept the Northeast in the spring.

“A bed is a gift right now,” said Jason Craig, regional chair for the Mayo Clinic Health System in northwest Wisconsin. “I’ll take all of them