Fighting back tears as he spoke, Metropolitan Police Commander Stuart Cundy said five people had been formally identified – but warned that some victims may never be identified.
The fifth victim has been officially named as 52-year-old Khadija Khalloufi.
“The awful reality is, as I’ve said before, is that due to the intensity of the fire and the devastation within Grenfell Tower that we may not be able to identify everybody that died,” he said.
“We have been from the top to the bottom of Grenfell Tower.
“The search operation will be painstaking and having been in there myself it’s really hard to describe the devastation that the fire has caused.”
Following his announcement, a minute’s silence was held across Britain to remember the victims of the fire, which ripped through the 24-storey London tower last week.
Several hundred people have rallied at a demonstration demanding “justice” for the victims.
Emma Dent Coad, the new Labour MP for Kensington, human rights barrister Michael Mansfield QC, and local residents were among the speakers at the event near the stricken block.
Sky’s Dan Whitehead said there is “anger” that although help is slowly being provided, some of those who were caught up in the blaze say there is a lack of co-ordination.
Meanwhile, a total of £202,000 of government money has been handed out to 180 families who were directly affected by the disaster.
The payments are being made up of £500 cash and £5,000 which is to go into bank accounts.
Mr Cundy said the death toll – which increased from 58 – could still change as investigations continue, but not as significantly as it has previously.
“I believe there may be people who were in Grenfell Tower that people may not know are missing and may not have realised they were in there on the night,” he said.
“Equally, there may be people who thankfully managed to escape the fire and for whatever reason have not let their family and friends know that they are safe.”
Mr Cundy said five people who were reported missing had now been found safe and well.
The latest figures from NHS England show that 17 patients are now being treated in four London hospitals. Of these, nine remain in critical care.
Mr Cundy said the search and recovery operation was still being carried out in the burnt-out tower.
“This is an incredible distressing time for families and they have my commitment that we will do this as quickly as we possibly can,” he said.
He added: “Sadly, for many families, they have lost more than one family member and my heart truly goes out to them.”
He said it had been “incredibly emotional working in there … On Saturday, I went in myself and went to the top floor.”
He promised an “exhaustive” criminal investigation that would also look at a major refurbishment of the tower completed last year.
“If I identify… an issue that is a risk to public safety, we will be sharing that immediately with the relevant authorities,” he said.
The Metropolitan Police has released footage showing the “indescribable” damage caused by the fire.