The Chancellor said a criminal investigation would examine whether building regulations had been violated when the block underwent an extensive renovation that was concluded a year ago. "So that 58 would include that 30", the spokesman said Saturday.
Sixteen bodies have been taken to the mortuary after flames tore through the 1970s tower block - while 14 others have been recovered from the building.
She told BBC One's Sunday Politics: "We are still hearing stories of people not being allocated properly".
The building's external cladding has come under intense scrutiny after the tragedy in which 58 people are presumed dead.
Opposition Labour Party lawmaker David Lammy said that the government and the police should immediately seize all documents relating to the building's renovation to prevent the destruction of evidence that could show criminal wrongdoing. Some said they had never seen a building fire advance so quickly.
"We've always been advocates of automatic fire sprinkler protection in tall tower blocks and they work - people don't die in sprinkler buildings", he said.
As the investigation began, Cundy urged missing residents who are alive and have not come forward to contact authorities to ensure their safety.
British health authorities say that 19 fire survivors are still being treated at London hospitals, and 10 of them remain in critical condition. Cundy said emergency workers have now reached the top of the building.
Prime Minister Theresa May, criticised for keeping her distance from angry residents during her visit to the charred remains of the 24-storey Grenfell Tower, said on Saturday the response to the disaster had been "not good enough".
Although police did not speculate on the eventual number of fatalities, local community sources say at least 70 from Grenfell Tower are still missing, including entire families. "He gave love to everyone", his family said in a statement.
The identification of the victims is proving very hard - which experts attribute to the extreme heat of the fire.More news: Ronaldo, Hernandez and VAR give the Confederations Cup the spectacle it needed
Nisha Parti, a film producer who has been helping organise help for victims of the fire in Kensington, revealed the local council is giving the measly sum to survivors when they are checked in to hotels. This relies on dental records, fingerprints and DNA when possible and also features like tattoos or scars.
The tragedy cast a pall on the Trooping the Color festivities that mark the official birthday of Queen Elizabeth II.
A minute's silence will be held at 11:00 am across Britain on Monday to remember the victims.
Media captionCouncil leader Nicholas Paget-Brown says the council has a "very well organised" operation in place.
"It is hard to escape a very somber national mood", Elizabeth said in a message on her official birthday. Hundreds have been left homeless, putting more pressure on officials in a city plagued by a chronic housing shortage.
The government has promised a full public inquiry, but that has done little to ease a sense of frustration at the lack of information about how the fire moved so quickly to engulf the building. British officials have ordered a review of other buildings that have had similar renovations.
"The response of the emergency services, NHS and the community has been heroic". May has announced a fund of about $6.5 million for the victims.
Many survivors are sleeping on the floor in community centers and there's still no coordinated distribution of donated food and clothing.
After a turbulent three months which has seen three militant attacks and now the tower blaze, Queen Elizabeth said "it is hard to escape a very sombre national mood", in a message on her official birthday.
Two nearby Underground lines were partially shut down Saturday in the fire area to make sure that debris did not land on the tracks.