Metropolitan Police Commander Stuart Cundy confirmed that the force now believes at least 79 people are dead or missing, presumed dead.
London police said Monday that 79 people were now believed to have died in the high-rise apartment building fire.
The blaze occurred early Wednesday and later engulfed all floors above the second in the 24-story building that contained about 120 apartments in north Kensington.
May said: "As we continue to respond to the needs of the community, our focus is on ensuring that all of those affected by this unimaginable tragedy get the right support as quickly as possible".
"Sadly, for many families, they have lost more than one family member, and my heart truly goes out to them", he said.
The revised death toll comes as images showing the burned-out interior of the London tower block are released.
Authorities have also faced questions about why the block, built in the 1970s and home to 125 families, was never fitted with a sprinkler system that might have saved lives.
"My understanding is the cladding in question, this flammable cladding which is banned in Europe and the United States, is also banned here", Hammond told the BBC earlier.
"It's hard to describe the devastation the fire has caused", Commander Cundy said, fighting back tears as he spoke.
The cause of Wednesday's blaze is still under investigation, but anger has mounted in the community amid reports that exterior paneling used in an extensive renovation completed previous year may have been banned by United Kingdom rules.
"Where offences have been committed, I will do everything in my gift to make sure they are brought to justice", he said.
The devastating fire erupted in the early hours of June 14 trapping several sleeping residents at the 24-storey Grenfell tower.
The material was captured by a specialist police recovery team which has been working closely alongside colleagues from the London Ambulance Service and the London Fire Brigade as part of the ongoing operation.
"Until we do that, we can not look at anything other than staying with the same advice, which has kept people safe in their buildings. This has not happened with these funds", they said.
Amid reports that the government is struggling to find housing for the hundreds of Grenfell Tower survivors left homeless by the blaze, Corbyn called on May to repossess empty homes and offer them to the displaced residents.
Over the past few days, another five people who had been reported missing in the fire have been found to be safe and well, Cundy said, saying he's "so grateful" for the news.