The Royal Wolverhampton NHS Trust has been asking that patients and visitors wear a face covering, if they can, at New Cross Hospital, West Park Hospital and Cannock Hospital since 11 July, to help keep patients and staff safe.
People are also urged to wear face coverings when they have GP appointments or are visiting any other healthcare setting, such as a care home, unless they have a reasonable reason not to do so.
Data released on Friday from the Office for National Statistics showed that a total of 3,760,200 people in private households in the UK were estimated to have had coronavirus in the week to 14 July – up from 3,498,700 people the previous week.
Councillor Jasbir Jaspal, the City of Wolverhampton Council's Cabinet Member for Public Health and Wellbeing, said: "We have seen a sustained rise in Covid-19 cases over recent weeks.
"So it's very important that we do all we can to try and stop the spread of the virus, and one way you can help is to make sure that you wear a face covering if you are attending a healthcare appointment or visiting a friend or relative in hospital or a residential care home.
“National guidance is also still in place which recommends wearing face coverings in crowded places, like in shops and on public transport, and I would encourage people to follow this guidance if they possibly can."
Dr Ananta Dave, Chief Medical Officer for the NHS Black Country Integrated Care Board, said: “Please make sure you wear a mask when you’re visiting your GP practice, or any other health care setting such as a hospital or care home, to help keep everyone safe, unless you have a valid exemption.”
Councillor Jaspal added: "Fortunately, despite high infection rates, the number of people falling seriously ill or dying with Covid-19 remains low at this time, and that's primarily due to the impact of the vaccine.
"So, if you haven't already had your first or second jab, or your booster, please get it as soon as possible. Remember that getting your booster when it is due is particularly important because we know that the effectiveness of the vaccine wanes after time.
"We have clinics operating throughout Wolverhampton, including a series of pop up clinics which will be opening over the next few weeks."
For details of local walk in Covid-19 vaccination clinics, please visit Find a walk-in coronavirus (COVID-19) vaccination site. People can also book an appointment via Book or manage a coronavirus (COVID-19) vaccination.