She also met with injured survivors in hospital and announced a £5 million ($6.4 million, 5.7 million euro) fund for emergency supplies, food and clothing.
When the prime minister visited the west London site of the fire on Thursday she spoke only to emergency services.
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 aluminium cladding, called Reynobond, is banned for buildings higher than 12m in the U.S., far lower than the 24-storey Grenfell Tower that was consumed by the roaring blaze, according to a salesman for the company that manufactures it.
The announcement by a spokesman comes a day after May was heckled during a visit to the west London neighborhood where Wednesday's inferno took place.
Later, demonstrators stormed the offices of Kensington and Chelsea Council over its handling of the crisis amid concerns that earlier renovation work was linked to the dramatic spread of the blaze.
For resident Soran Karimi, 31, who lives in the block opposite, it was nothing short of "murder" and "people should be prosecuted for this".
Search-and-recovery efforts were paused on Friday over safety concerns, but resumed Saturday afternoon, police said.
He said there may have been people in the tower that police are not aware of, which would add to the death toll.
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The council said 110 households had been given temporary accommodation by Friday morning, and added that it was working to find more permanent homes.
Questions remain over the fire as it emerged there were no sprinklers or communal alarm in Grenfell Tower.
Mrs May has pledged a £5 million support package for the victims, which included a promise to rehouse them within three weeks.
Police have warned some of the victims may never be identified due to the state of the remains.
He also formally confirmed named the first fire victims as Mohammed al-Haj Ali, a 23-year-old tower resident.
She has now ordered daily updates on the progress of rehousing victims and has deployed more staff across north Kensington and on phone lines to support victims.
"It might be that some of those are safe and well, but for whatever reason have not wished to let us know".
A message from The Queen on Her Majesty's Official Birthday. They are furious that people still are missing and that the local government hasn't done enough to help them in the aftermath.
"We will follow the recommendations of the public inquiry", he said.
"We've already started to get a lot of people calling in to say, "Of course we will do it"."Some jumped to their deaths rather than face the flames, and witnesses reported seeing small children thrown from the tower by their families in a desperate bid to save them.