As she left St Clement's Church following a visit lasting less than an hour, the PM faced cries of "coward" and "shame on you". "This number 58 may change".
"It really devastated me", he said.
"I have heard the concerns and I have ordered immediate action across the board to help victims' relatives and the survivors".
May announced on Friday that her government has set up a £5 million ($8.4 million Cdn) "discretionary fund" to help pay for food, clothing, personal supplies and funerals.
May also said phone lines would be better staffed and more staff would be deployed in the area.
"We need answers and we need answers now", another man said through a megaphone.
Two of the dead have been confirmed as 24-year-old artist Khadija Saye and Syrian refugee Mohammed Alhajali (23).
The identification of the victims is proving very hard - which experts attribute to the extreme heat of the fire.
Protestors gather outside Downing Street.
Rescue personnel have had difficulty reaching the top floors of the charred, 24-story tower.
In a television interview Friday, she sidestepped questions over whether she had misread the public mood.
LeT leader, 2 civilians among 14 killed in Kashmir unrest
The victims were travelling in a police jeep on the Anantnag-Achabal road when the incident occurred. The militants started indiscriminate firing at the police party using their assault rifles.
She says she will receive daily reports from the stricken neighborhood, where hundreds of people have been displaced.
Anger is rising against British authorities over safety failings and a widely perceived slow release of information. This year, however, it is hard to escape a very sombre national mood.
Police said Saturday that their investigation would look at the building and its 2016 refurbishment, and vowed to bring prosecutions "if there is evidence". The Telegraph collected some of the known names of the dead with pictures, mostly from the accounts of people searching for lost friends and family members.
"They haven't got easy fire escapes, they've got no sprinklers, it's totally, totally unacceptable in Britain that this is allowed to happen and people lose their lives in this way".
The tragedy has provoked a huge response from nearby communities.
Three appeals on the JustGiving site have helped to raise the 3 million pounds, and London's Evening Standard newspaper has launched a separate appeal that has raised at least 1.5 million pounds ($1.9 million) by Saturday morning. Like many other residents she has spent the days following the fire living in temporary shelters with minimal sleep, printing and distributing posters wherever she can.
He said: "The conditions due to the fire damage verge on indescribable, which is why this will be such a lengthy operation taking weeks to complete".
The Chancellor added: "When the inquiry produces its recommendations we will act on them".
This latest figure, based on reports from the public, included the 30 already confirmed to have died in the fire in west London early Wednesday.