Sixteen bodies have been removed from the blackened 24-storey apartment block, and the first victim was formally identified as Mohammed Alhajali, a 23-year-old Syrian refugee.
Queen Elizabeth II said the disaster had cast a sombre pall over Britain, but insisted the country was showing resolve in the face of adversity.
However, public anger has swollen over the disaster, with furious residents heckling May and storming the local authority headquarters on Friday.
They demanded justice for the victims and claimed Wednesday’s fatal blaze was due to negligence, with many citing the cladding that was put on the 1974 concrete tower during a recent refit.
“It was a death trap, and they knew it,” one person shouted as demonstrators stormed inside the offices of the Kensington and Chelsea council, which was responsible for managing the social housing block in a working-class enclave of one of Britain’s richest districts.
Police said that their investigation into the blaze would look at the building and its 2016 refurbishment, and vowed to bring prosecutions “if there is evidence”.
“There are 58 people who we have been told were in Grenfell Tower on the night that are missing and therefore, sadly, I have to assume that they are dead,” police commander Stuart Cundy told reporters at the scene.
He said the number may vary should further information come to light, and tried to field questions from restive locals.
The area surrounding the tower has been plastered by distraught relatives with pictures of the missing, from grandparents to young children.
Seven Moroccans are among the dead, the north African country’s foreign affairs ministry said.
May, who has faced accusations of misreading the public mood, met a 15-strong group of victims, residents, volunteers and community leaders at her Downing Street office.
The meeting happened as a wide-ranging demonstration took place outside, protesting about several issues including the Grenfell Tower fire.
The group, most dressed in casual clothes, made clear their demands to May in a two-and-a-half-hour meeting.
A man from the delegation, who did not give his name, told reporters they would make a full statement “in the community”.
He said the group had spoken about their “demands and what we expect”.
May chaired a government task force on the disaster before meeting survivors.
19 patients are still being treated in hospital, of whom 10 are in a critical condition, the National Health Service said Saturday.
The emergency services expect to find no more survivors.
May was criticised for avoiding locals when she visited the scene on Thursday and faced cries of “Shame on you” and “coward” when she returned the following day, with police breaking up scuffles.
Queen Elizabeth and her grandson Prince William visited a community centre on Friday where some survivors are being housed, and where volunteers have been inundated with donations of clothes and food.
The head of state said a saddened country was showing a determination to rebuild lives wrecked by “terrible” tragedy.
She stood for a minute’s silence before her 91st birthday parade on Saturday.
“It is difficult to escape a very sombre national mood,” she said in a message marking the event.
There are questions about why the block was not fitted with sprinklers or a central smoke alarm, and whether a recent refurbishment, including new external cladding, helped fuel the flames.
The fire forced residents to flee through black smoke down the single stairwell, jump out of windows or even drop their children from the 120-apartment building.
Saturday’s newspapers conveyed the sense of anger on the streets.
“It was murder” said the front page of The Sun; “May takes cover” said The Times; “Inferno: the anger erupts”, said the Daily Mail.
The Daily Telegraph claimed left-wing militants had been hijacking the demonstrations and deliberately spreading political fake news conspiracy theories about the blaze.
May has announced a judge-led inquiry into what happened, and on Friday promised £5 million ($6.4 million, 5.7 million euros) for emergency supplies, food and clothing.
First Secretary of State Damian Green, May’s deputy, told BBC radio that the inquiry would have interim reports and “we want the response to be as fast as possible”.
Green said the government would pay for residents’ legal representation at the inquiry and reiterated May’s promise to rehouse those displaced by the fire within three weeks, as close as possible to home.