The former attorney general described the proposed prime minister's agreement to the European Union as "fairly second-rate."
MP Tory was among the roughly 700,000 people who participated in the popular vote march through London on Saturday, in what had been billed as the largest demonstration against Brexit to date.
Grieve told the BBC Radio 4 Today program: "If we do not do this now, we will lose the opportunity and once we leave the EU on March 29, going back in is completely different than deciding to stay.
"My opinion has grown in recent months that, in fact, we are reaching a dead end and that there could be some kind of agreement on offer, but I have to say that I think it is quite important.
"When this type of problem occurs, in view of the first referendum, the only solution is to go back to the public and say that this is what you really want. And if they want it, so be it.
Deputy Remainer said that the EU referendum in June 2016 was based on an "abstract discussion" and warned that Britain is on track to deliver a Brexit that "very few people want".
He added: "I think this is the end, if people discover that what they are about to obtain is what they want, then I, as a democrat, would accept it."
"But what worries me is that I think we are about to deliver a Brexit that, in fact, very few people want."
After another stalemate in talks with EU leaders in Brussels, Theresa May said she would consider extending the transition period beyond December 2020 to allow the UK and the EU to denounce the problems surrounding the EU. the future post-Brexit relationship, above all, the problem of the Irish border.
But this movement has provoked a violent reaction from Tory's parliamentarians, with Archbishop Jacob Rees-Mogg dismissing it as "a rather bad attempt to kick the can along the way."
Mr. Grieve admitted that it was likely that the transition period would "extend over three years as we try to break the relationship continuously."
He added: "That shows me why, in fact, we're embarked on the project, which is totally different from what people might have some reason to expect in 2016.
"We are in the most extraordinary situation in which we remain tied to all the EU rules without any ability to influence the contribution."
On Friday, Jeremy Hunt urged conservative at war to endorse Ms. May's Brexit strategy while continuing to face a backlash against the possibility of extending the transition period.
The foreign secretary said that an EU proposal to continue with the implementation process "could help" to reach an agreement on the future relationship and rejected claims that it amounted to "capitulation".
Speaking on the BBC Radio 4 Today show, he said: "The reason why this week has been difficult is because Theresa May has not given up."
"He has remained firm, he has remained true to his principles.
"It is precisely because he has not capitulated that we have not concluded this agreement.
"The only thing I would say to my colleagues is that the great strength of the EU in these negotiations is that the 27 EU nations have held together.
"Now we must do the same behind Theresa May to maximize her bargaining influence in Brussels and ensure that she returns with that agreement that honors the letter and spirit of the referendum decision."
But the EU's chief negotiator, Michel Barnier, warned that the UK withdrawal agreement could still fail on the issue of the Irish border, which would lead to a Brexit without "extremely serious" agreements.
Speaking to France-Inter radio, he said the agreement had been completed by 90 percent and that he hoped to complete it "in a few weeks or a few months, as soon as possible."
But when asked if he was convinced of reaching an agreement, he replied: "I do not have a deep conviction on this issue, because in the United Kingdom the political situation is very complex and I do not know what decisions Theresa May will make."