There has been a 14% rise in people testing positive in the week to 20 September – the biggest increase since the summer.
But there is no clear evidence of an autumn Covid wave starting, says the Office for National Statistics (ONS).
More recent data showing a rise in hospital admissions with Covid has been called “a wake-up call”.
Daily hospital admissions are lower than where they were for much of July, but highest among the oldest age groups.
However, six out of 10 people with Covid in hospital is being treated for something else – not Covid-19.
“The fact there are people getting so seriously ill they need to go into hospital is a wake-up call to us all that Covid is still here,” said Dr Waite.
Health experts have warned of a flu and Covid “twindemic” this winter, urging those who qualify to get their free jabs now.
Although Covid is increasing in England and Wales, the trend is uncertain in Scotland and Northern Ireland, the ONS says.
Sarah Crofts, from the ONS Covid-19 Infection Survey, said: “It is too early to identify whether this is the start of a new wave of infections. We will continue to closely monitor the data.”
The ONS estimates are based on thousands of random tests on people in private households across the UK, whether or not they have symptoms.
The most common Covid symptom is currently a sore throat, with fever and loss of smell much rarer, according to symptoms logged by 3,000 people testing positive via the Covid symptom app.
Booster jabs against Covid, alongside flu vaccinations, are now being offered to the most vulnerable to help protection over the winter.
Most people will receive a new type of vaccine made by Pfizer or Moderna – which tackles both the original Covid virus and the recent Omicron variant, offering better protection.