In a memo to staff, Walmart said it would stop offering paid Covid-19-related sick leave next month, except when required by local governments. The policy gave hourly workers extra paid time off for Covid-19 illnesses, exposures or Covid-like symptoms, beyond their standard sick-leave pay.

It will also drop a daily health screening given to workers when they start a shift in all areas except a handful of states that require it. Unvaccinated workers must still wear masks, the memo said.

Walmart was one of the first big private employers to adopt a mask requirement in 2020 when the Covid-19 pandemic emerged. Walmart eased its mask requirements for vaccinated workers and shoppers in May 2021 but then reimposed them for some workers last July when the Delta variant caused a jump in infections. It then required masks for all workers last December because of the Omicron variant.

The new policies come after officials in New York, California, Illinois and several other states said this week they would lift mask mandates as cases from the Omicron variant drop. School districts around the country are also rolling back Covid-19 policies.

On Thursday, rival

Amazon.com Inc.

said it would allow fully vaccinated employees to go maskless inside its warehouses and that it was taking away Covid-19-related paid leave for employees who aren’t vaccinated.

Local governments and large companies are dropping mask mandates before similar changes from the Centers for Disease Control and Prevention. Since the pandemic began, most large retailers have moved in lockstep with the CDC on vaccine and mask policies.

CDC officials said they are optimistic about falling Covid-19 rates and are working on new guidance around masking but that the agency still recommends the use of masks in indoor public places, including schools.

Write to Sarah Nassauer at sarah.nassauer@wsj.com