"I have ordered that more staff be deployed across the area, wearing high visibility clothing, so they can easily be found, dispense advice and ensure the right support is provided", May said after a meeting with survivors at her Downing Street office on Saturday.
Metropolitan Police commander Stuart Cundy said: "That number 58 may change".
The British government on Sunday vowed to act on any recommendations from an inquest into the a fire that consumed a 24-story high-rise tower in the west of London and killed at least 58 people. "I really hope it won't, but it may increase".
The block, which housed some 600 people in about 120 apartments, is in a low-rent housing estate next to one of the most affluent areas of Britain.
"We have colleagues in there as we speak, searching for and recovering those that have died".
Under pressure from the interviewer Mark Mardell, he later said: "This was a huge, sudden disaster, a complete tragedy and no one borough alone would be able to cope with the scale of it".
He said police had received some 70 pictures and videos of the fire from the public and urged them to send more as officers investigate the blaze.
Residents demanded that Cundy explain why more information had not been released on the numbers missing and why more identities had not been confirmed.
Earlier on Saturday, Mrs May chaired a Government task force on the disaster.
Cundy gave Alhajali's name at the press briefing.
She has promised those left homeless would be rehoused locally within weeks, and the £5m fund would pay for emergency supplies, food, clothes and other costs.
Mr Magee and his crew avoided falling debris as they battled their way in to the building, only to find conditions worse inside.
Around 70 people are missing, according to Britain's Press Association, and identification of the victims is proving very hard.
Kate Taylor, who lives about 3 miles from the tower, said she joined the protest to show solidarity with the victims, and said she agreed with the shouts of "Tories out".
The sister of a man still missing in the London blaze told reporters that when she phoned him on the 21st floor as the fire spread, he said he hadn't evacuated with his wife and three children because fire officials told him to "stay inside, stay in one room together and put towels under the door".
A woman touches a poster for 12-year-old Jessica Urbano on a tribute wall Friday after laying flowers on the side of a church next to the Grenfell Tower in London.
"We've already started to get a lot of people calling in to say, "Of course we will do it"."
"We were aware the place was filling up with more and more smoke, we were not supposed to be starting off in smoky environments but there wasn't a safe zone to go to".
Mr Cundy said: "The investigation will be exhaustive. They feel the Government and local council haven't done enough to help them in the aftermath of this horrific incident, or to provide answers to their increasingly urgent questions".