TEXAS — Multiple Texas counties and cities are ordering businesses to require customers and workers to wear face masks as the state sees a continued rise in the numbers of new coronavirus cases and hospitalizations.

Republican Gov. Greg Abbott, who has refused to order a face covering mandate for individuals, said this week that local governments can instead order businesses to require them.

Bexar County was the first, and local officials in some of the state’s most populous areas, including the city of Austin, Dallas County, Harris County and El Paso County, quickly adopted similar measures with fines ranging from $500 to $1,000 for businesses that don’t comply.

Texas health officials on Friday reported 3,454 new cases, a slight dip from Thursday’s record high of 3,516. But Texas also reported a new record of 3,148 hospitalizations, more than double the total on Memorial Day.

The actual number of people who have contracted the virus that causes COVID-19 is likely higher because many people have not been tested and studies suggest that people can be infected and not feel sick.

“The virus is here. Infections are rising. Hospital capacity is filling up. If we want the economy to reopen fully and stay open, we have to take this seriously,” Austin Mayor Steve Adler tweeted Friday.

The local orders have been criticized by small business advocates who complain they are turning shop owners and retailers into “mask police.” Some conservative lawmakers blame Abbott for giving his blessing on such measures.

“Texas has now gone full circle from a dictatorship to a republic, to a sovereign American State. Now it appears that as long as we allow the Governor’s actions ... we are expected to live as if we have a monarchy,” Republican state Sen. Bob Hall said Friday in a statement posted on his website.

Abbott’s aggressive push to reopen the economy continued Friday with amusement parks and carnivals around the state allowed to reopen. Six Flags parks in Arlington and San Antonio opened to members and pass holders and will open to everyone on Monday.

The Texas Alcoholic Beverage Commission conducted undercover inspections of bars this week to see if they were following social distancing guidelines and found most were in compliance. Of more than 200 inspected, only three —in Dallas, McAllen and El Paso— had their alcohol licenses suspended. An agency spokesman said inspections will continue this weekend.

Texas is set to unveil guidelines for a return to the school in August for the fall semester. Education Commissioner Mike Morath said this week that state officials have determined it will be safe for public schools to reopen. The state will not require more than 5 million Texas students to wear masks, but districts will have flexibility to set local policies.

The state also will provide some flexibility for families who want to keep their children in distance learning programs from home, Morath said.

Abbott ordered schools to close March 19, and teacher advocates question whether the reopening will adequately address safety. The Texas American Federation of Teachers and the Texas State Teachers Association both attacked moves to reopen as unsafe.

For most people, the coronavirus causes mild or moderate symptoms that clear up within weeks. But for others, especially older adults and people with existing health problems, the highly contagious virus can cause severe illness, including pneumonia, and be fatal.

The Texas Workforce Commission announced Friday the state added 291,000 private sector jobs in May, reducing the state’s jobless rate during the pandemic from a record high 14% in April to 13%. Most of the jobs came in the leisure and hospitality industry, the commission said.