Tony Blair has said it is his "mission" to persuade Britons to "rise up" and change their minds on Brexit.
Speaking in the City of London, the former prime minister claimed that people voted in the referendum "without knowledge of the true terms of Brexit".
He urged "a way out from the present rush over the cliff's edge".
Former Tory leader Iain Duncan Smith said the comments were arrogant and undemocratic but Lib Dem Nick Clegg said he "agreed with every word".
Former UKIP leader Nigel Farage said Mr Blair was "yesterday's man" while Downing Street said it was "absolutely committed" to seeing Brexit through.
Foreign Secretary Boris Johnson added: "I urge the British people to rise up and turn off the TV next time Blair comes on with his condescending campaign."
Prime Minister Theresa May wants to trigger formal Brexit talks by the end of March - a move which was backed in the House of Commons by MPs last week.
'Opportunity to reconsider'
Mr Blair, who was UK prime minister between 1997 and 2007, used the speech to the pro-European campaign group Open Britain to argue that leaving the EU would be "painful" for Britain and Europe and the benefits would be "largely illusory".
Mr Blair, who campaigned to remain in the EU, said that while he accepted that people voted to leave by 52% to 48%, he would recommend looking again at Brexit when "we have a clear sense of where we're going".
Pressed on whether he thought there should be a second referendum, he said: "All I'm saying is a very, very simple thing, that this is the beginning of the debate - that if a significant part of that 52% show real change of mind, however you measure it, we should have the opportunity to reconsider this decision.
"Whether you do it through another referendum or another method, that's a second order question.
"But this issue is the single most important decision this country has taken since the Second World War and debate can't now be shut down about it."
Analysis by political correspondent Tom Bateman
Tony Blair's warnings about the risks of Brexit might have made some viewers believe the referendum campaign was still being fought.
But his central political point takes us onto new ground - that the voters could still change their minds about leaving the EU and Remainers should persuade them to do so.
It will be seen by some as a call to arms - Tony Blair's Brexit insurrection.
Brexiteer MPs were unsurprisingly excoriating, with the foreign secretary hinting at what Mr Blair's opponents see as his toxicity after the Iraq war.
But importantly the former PM's speech raises a tactical question for Remainer MPs wondering what to do next: fight for Brexit on their terms or fight Brexit itself.
In the absence of an effective opposition, he said pro-Europeans needed to build a "movement " reaching across party lines, he said, adding the institute he is launching would play its part in developing the arguments to rethink the country's position.
"The debilitation of the Labour Party is the facilitator of Brexit. I hate to say that, but it is true."
'Pain and gain'
While he fully accepted immigration was "a substantial issue", he said it had become the "primary consideration" for the government and suggested the public were more concerned about arrivals from outside the EU.
Mr Blair has faced criticism in the past for his government's decision to allow people from Poland, Hungary and the Czech Republic to work in Britain without restrictions, while most EU states imposed transitional controls to slow the rate of migration.
Mr Blair stressed that the Conservative government only "has bandwidth for only one thing - Brexit", at the cost of the NHS, education, investment in communities, the rise in serious crime, the increased burden of social care and control of immigration.
"This is a government for Brexit, of Brexit and dominated by Brexit, he said, adding that the issue was the government's "waking thought, the daily grind, the meditation before sleep and the stuff of its dreams or nightmares".
Iain Duncan Smith, who was a prominent Leave campaigner, said Mr Blair had shown the political elite was completely out of touch with the British people.
He compared Mr Blair returning to the political scene to the British horror comedy "Shaun of the Dead", with "his hands outstretched to tell the British people they were too stupid to be able to understand what they were voting on", adding that this "is both arrogant and a form of bullying".
And Mr Farage described Mr Blair as a "former heavyweight champion coming out of retirement" who would "end up on the canvas".
Kate Hoey, a prominent Leave campaigner and Labour former minister, told the BBC she did not think anyone would take Mr Blair's "patronising" opinion seriously.
"I'm really quite sad that he doesn't feel that as a former prime minister - he's travelled all round the world, he's made himself lots of money - he's come back. Why doesn't he just now go and find himself a job?"
But Alan Johnson, who led Labour's campaign to keep Britain in the EU - urged people to listen to the message, not the messenger.
Stressing he would not rule out a second referendum, Mr Johnson said people are concerned that Britain could end up as a "low tax, anything goes, race-to-the-bottom kind of country" post Brexit.
Supporters of leaving the EU argue it will free up the UK to trade better globally and give the government better control of immigration.
Earlier this month, MPs overwhelmingly agreed, by 494 votes to 122, to let the government begin the UK's departure from the EU by voting for the Brexit bill.
The Commons vote prompted splits in the Labour party. Despite calls by leader Jeremy Corbyn for his party to back the government, 52 MPs rebelled.
Lib Dem attempts to amend the bill to include a provision for another referendum were defeated by 340 votes to 33.