Even with the grudging support of party leader Jeremy Corbyn, it is far from likely that another plebiscite will enjoy a majority in Parliament - but the option has become less theoretical.
The PM said she was focused on leaving the European Union with a deal on 29 March.
Lawmakers are nearly certain to reject such an option and will then have a March 14 vote on a short Brexit delay, probably extending until late June.
With just 30 days to go before Britain is scheduled to leave the European Union, the Parliament endorsed Prime Minister Theresa May's concession that if the MPs can not agree on a deal to sever ties with the continent, Brexit might have to be delayed.
European Chief Negotiator Michel Barnier said on Thursday that the United Kingdom needed to make a decision rather than call for a possible postponement of the Brexit deadline, adding that the talks on the United Kingdom withdrawal from Europe were at "a grave moment".
"If we try to stay and we stay beyond the European elections, there will only be one victor from that, and that would be Tommy Robinson", he said in comments reported by The Telegraph.
But Brexiteers in her own party mounted a show of strength in votes on Wednesday night, with over 100 Tory MPs breaking a three-line whip to oppose the move.
She said: "I agree with you that Parliament must now come together and honour the referendum result by voting for a deal which will give businesses and citizens the certainty they need and deserve".
Jackets Acquire Duchene For Prospects
Kekalainen added: "I'm not going to get into any specifics or details, but it's not accurate that we haven't had an offer". They acquired Chris Wideman from Florida, and Erik Gudbranson in a deal that sent Tanner Pearson to Vancouver.
A new party backed by Nigel Farage, the insurgent who helped shove Britain toward the European Union exit, has a message for the country's leaders: The foundations of the political system will explode if Brexit is betrayed. It won't contradict the deal but will aim to allow British Attorney General Geoffrey Cox to change his legal advice, which previously said that the United Kingdom could be trapped permanently in the "backstop" arrangement that prevents a hard Irish border.
French President Emmanuel Macron, speaking in a joint news conference with German Chancellor Angela Merkel, said a Brexit delay would need to be "justified" by "new British choices", suggesting that May's Government would need to signal where this is all headed - versus asking for more time just because they are stuck.
Theresa May has called calls on MPs to "do their duty" and vote through her deal to end the Brexit deadlock.
But May's climbdown has taken much of the sting out of the occasion, with no resignations expected and the amendment's tabler Dame Caroline Spelman indicating she may not force it to a vote if she receives adequate reassurance on the PM's intentions.
May has already said she will protect the rights of those in Britain, but one of her ministers has now agreed to write to European Union institutions to urge a joint approach.
Before Mrs May's concessions on Tuesday, Ms Cooper had been assembling support from all parties for an amendment that would have ensured Parliament had a chance to block a no-deal exit and seek a delay.
If they refuse, May 14 will put forward a proposal to postpone Brexit.
Business leaders are triggering contingency plans to cope with additional checks on the post-Brexit UK-EU border they fear will clog ports, silt up the arteries of trade and dislocate supply chains in Europe and beyond.