The figures come as Health Secretary Matt Hancock indicated ministers at Westminster were open to delaying the easing of restrictions in England scheduled for June 21, while the Scottish Government has paused a relaxing of rules in parts of central Scotland.
The rise in rates has yet to be mirrored by a steady increase in Covid-19 hospital cases, however, with the latest data showing patient numbers have climbed slightly to levels last seen at the end of May.
Nearly three-quarters of local areas of the UK (283 out of 380) recorded a week-on-week rise in Covid-19 case rates for the seven days to June 2.
This is the highest proportion since January 6.
In just under half of these places (134), the rise was small enough to be in single figures.
But a growing number of areas are recording sharp jumps in rates, some of them large enough to push them above the symbolic level of 100 cases per 100,000 people.
When using a seven-day average to flatten out fluctuations in the data, 256 of 380 local areas in the UK are currently recording a rise in rates – the highest number since January 10.
The analysis has been compiled by the PA news agency using Public Health England data published on Sunday on the Government’s coronavirus dashboard.
Data for the most recent four days (June 3-6) has been excluded as it is incomplete and does not reflect the true number of cases.
Blackburn with Darwen in Lancashire continues to record the highest rate in the UK, with 515.7 cases per 100,000 people in the seven days to June 2, up from 390.8 a week earlier.
It is one of 28 local authorities currently recording rates above 100 cases per 100,000 – the highest number since March 28.
Other areas that are now above this threshold include Stockport (up week-on-week from 50.4 to 146.2), Trafford (54.8 to 117.5) and Wigan (66.6 to 116.5) in Greater Manchester, along with Chorley (79.5 to 122.7), Preston (95.0 to 169.8) and Pendle (72.7 to 107.5) in Lancashire.
Eight of the 10 areas in Greater Manchester are currently above 100 cases per 100,000 – though one of them, Bolton, is continuing to record a fall in rates, and is currently down week-on-week from 390.5 to 318.2.
Outside north-west England, sharp increases have been recorded in places including Staffordshire Moorlands (from 5.1 to 81.3), Bracknell Forest in Berkshire (20.4 to 94.7) and Three Rivers in Hertfordshire (15.0 to 71.8).
In Scotland, steep jumps have been recorded in South Ayrshire (from 46.2 to 158.1), Edinburgh (65.9 to 134.3) and Angus (21.5 to 87.8).
The rise in rates in these and other areas is being driven partly by surge testing to detect cases of the Covid-19 variant that originated in India, also known as the Delta variant.
But separate figures published on Friday by the Office for National Statistics (ONS) suggest infection levels are now rising more generally in England, with around one in 640 people in private households estimated to have had Covid-19 in the week to May 29 – up from one in 1,120 in the previous week.
Across England, three-quarters of local areas are currently seeing a rise in rates, including 30 of the 32 local authority areas in London and 35 out of 39 in north-west England, while in Scotland rates are up in 31 of the 32 council areas.
The trend is less clear in the other two nations, with with half of the local areas in Wales (11 of 22) recording a slight rise in the latest figures, together with five of the 11 areas in Northern Ireland.
The UK’s overall rate of new cases stood at 41.6 per 100,000 people in the seven days to June 2 – the highest since 43.9 on March 31.
But this is still well below the peak of the second wave, when the UK’s rate rose as high as 641.8 on January 4.
Pressed on the BBC’s Andrew Marr Show on Sunday over whether June 21’s easing of restrictions in England could be postponed, Health Secretary Matt Hancock said: “We are absolutely open to doing that if that’s what needs to happen.”
He suggested that the Indian (Delta) variant was up to 40% more transmissible than the strain first identified in Kent (Alpha) – but that vaccines were “working” and that “very few” people who had been fully vaccinated were ending up in hospital.
“The vaccine is working and it is our way through it, but the new variant, because it’s more transmissible, is obviously making that calculation harder and we’ll look at the data for another week and then make a judgment,” Mr Hancock said.
There were 932 patients in hospital in the UK with Covid-19 on June 3, up from 867 a week earlier.
The seven-day average for patients in hospital stood at 912 on June 3 – the highest number since May 26.