Ministers admit they expect to see cases rise further before the wave of the Delta variant starts to dip.
New figures reveal that all areas of the West Midlands, Shropshire and Staffordshire are now seeing spiralling cases.
Deaths and hospitalisations have shown a slight rise, but nowhere near the levels experienced in previous outbreaks.
Of the 315 local areas in England, 311 have seen a week-on-week rise in rates and only four have seen a fall.
South Tyneside in the North East has the highest rate, with 1,156 new cases in the seven days to July 2, the equivalent of 765.7 per 100,000 people.
This is up from 282.2 in the seven days to June 25.
Tamworth in Staffordshire has the second highest rate, up from 319.4 to 751.0, with 576 new cases.
The Government strategy appears to be to continue vaccinating so that all adults are double jabbed by around September – by which time the third wave should have declined.
Around nine in 10 adults in most parts of the UK are now likely to have Covid-19 antibodies, new figures suggest. The estimates range from 89.8 per cent in England and 91.8 per cent in Wales. In Scotland the estimate is lower and is closer to eight in 10 adults, or 84.7 per cent.
The presence of coronavirus antibodies suggests someone has had the infection in the past or has been vaccinated.
It takes between two and three weeks after infection or vaccination for the body to make enough antibodies to fight the virus. Antibodies then remain in the blood at low levels, although these levels can decline over time to the point that tests can no longer detect them.
The latest estimates are from the Office for National Statistics (ONS) and are based on a sample of blood test results for the week beginning June 14.
The estimates are for people in private households and do not include settings such as hospitals and care homes. In England, the latest estimate of 89.8 per cent adults is up from 79.6 per cent a month ago, while in Wales the estimate of 91.8 per cent is up from 82.1 per cent.
The ONS said there is a clear pattern between vaccination and testing positive for Covid-19 antibodies but “the detection of antibodies alone is not a precise measure of the immunity protection given by vaccination”.
Once infected or vaccinated, the length of time antibodies remain at detectable levels in the blood is not fully known.