While he pleaded with Arizonans on Monday to “mask up,” so far Mr. Ducey has refrained from issuing a statewide mask order or reissuing the stay-at-home order he rolled back beginning in mid-May.

“Our expectation is that our numbers will be worse” next week, Mr. Ducey said at a news conference in Phoenix. “We’re going to continue to adjust based on what the data or the information says.”

Even Monday’s executive order allows individual jurisdictions in the state to stage events with large crowds, provided they implement social distancing measures.

Mr. Ducey issued the new executive order under intense pressure, after weeks of videos and social media posts showed scores of people without masks partying in Scottsdale bars, and over the weekend, in inner tubes on a river. Mr. Ducey called Monday’s order, to be enforced by public health officials, “the most severe possible enforcement we have in our tool kit.”

Still, it was unclear on Monday exactly how enforcing the order would work, since it is silent on many other activities, such as youth sports. “We want to provide some flexibility,” Mr. Ducey said. “Personal responsibility and responsible behavior can go a long way.”

In Florida, where daily case counts soared to new records over the weekend, the city of Jacksonville said Monday that face masks would be required in any indoor public place where social distancing was not possible. The city is scheduled to host the Republican National Convention in August.

Several other cities and counties across the state are also moving to reimpose restrictions in response to the surge, which has followed the reopening of beaches, bars, restaurants and other social activities. South Florida counties closed their beaches for the Fourth of July holiday weekend.

Gov. Bill Lee of Tennessee signed an executive order on Monday extending his state’s emergency order and some other pandemic measures until Aug. 29. Tennessee recorded a record number of new cases on Monday.

Gov. Brian Kemp of Georgia is expected to extend for another two weeks the state’s remaining pandemic restrictions, which are set to expire on Wednesday, The Atlanta Journal-Constitution reported.

In Kansas, Gov. Laura Kelly said she would sign an executive order requiring most residents to wear masks in public beginning July 3.

“This is a simple, proactive step we can take to keep Kansans at work, get our kids back to school, and keep ourselves and our neighbors healthy,” she said in a statement. “Wearing a mask is not only safe — but it is necessary to avoid another shut down.”

The governors of New York and New Jersey said on Monday that they were so alarmed by a surge in coronavirus cases in the South and West that they were reconsidering plans to begin allowing indoor dining in restaurants in their own states. Gov. Philip D. Murphy of New Jersey said the step, which had been scheduled for Thursday, would be put off “indefinitely.” Gov. Andrew M. Cuomo of New York said officials would make a decision on Wednesday on whether to proceed.

Case counts have climbed sharply in many of the states that were first to reopen, including Florida and Texas, which recently forced bars to close again.