Campaigners for a second Brexit referendum are hoping to harness the energy demonstrated by Saturday’s 700,000-strong march in London to lobby MPs in the run-up to a crucial Commons vote.

The People’s Vote group will focus on 50 Conservative MPs – including five ministers – who they believe could be persuaded to vote for a second referendum should Theresa May’s final Brexit deal be voted down.

About eight Tory MPs, including Anna Soubry and Dominic Grieve, have publicly declared their support for another referendum. However, People’s Vote sources say there are a dozen more who are “on the doorstep” and a further 30 who are “reachable”, though they will not name them.

Alastair Campbell, a former No 10 director of communications under Tony Blair, emailed People’s Vote supporters on Sunday urging them to show the “same passion and urgency” they did on the march in lobbying MPs.

“I come from a Labour background but we’re working across the political spectrum with MPs and activists from all parties in a non-stop effort to win this fight,” Campbell wrote.

However, other remain campaigners question whether it would be possible to win over more than two dozen Tories, and argue that it would be better to focus on winning the support of Labour MPs in leave-voting areas, such as Gareth Snell and Ruth Smeeth, who represent constituencies in Stoke-on-Trent.

Grassroots activists have been encouraged to lobby key MPs and hold events at their constituency surgeries. People’s Vote says 30,000 supporters have indicated they are willing to help.

On Saturday, hundreds of thousands of people marched through the centre of London to Westminster in the biggest public protest against a government policy since the decision to go to war in Iraq.

A hundred thousand postcards were distributed as part of a campaign – #Writethiswrong – urging people to send a message to their MP in support of a second referendum.

Parliament would have to authorise any second national vote through primary legislation, and there are differences of opinion on the best time to push for a vote in the Commons on the principle of holding one.

The Conservative MP Sarah Wollaston speaks at the People’s Vote march. Photograph: Michael Bowles/REX/Shutterstock

To succeed, a second referendum would need the support of at least 15 Tory MPs, though this number would need to rise if more than a handful of Labour MPs voted against it.

“The best opportunity for winning over a lot of Tories could be to hold a vote after the Commons votes down Theresa May’s final deal, not before,” a People’s Vote source said. The former home secretary Amber Rudd has said she would favour a second referendum over a no-deal scenario.

Labour’s policy is to insist that May’s final Brexit deal protects workers’ rights and environmental standards as well as keeping the UK in a customs union with the EU. If those tests are not met the party has said it would vote down May’s final deal, in the hope of forcing a general election. Failing that it has said “all options are on the table” – including a second referendum.

Sir Keir Starmer, the shadow Brexit secretary, described Saturday’s march as significant and argued that both leave and remain supporters were “utterly losing confidence in the prime minister, and that reflects a much bigger concern about where this is going”.

Apearing on the BBC’s Andrew Marr show on Sunday, Starmer reiterated that if May’s Brexit deal was voted down then one of the options would be another “public vote” in which “remain has to be an option”.

People’s Vote will pay for social media advertising targeted at key MPs and their constituents, while a “six-figure sum” has been allocated to conduct polling in individual constituencies to try to demonstrate where an MP may be out of step with local opinion.

Best for Britain, another group campaigning for a second referendum, said it would work with campaigners to lobby Tory MPs. A spokesman said: “We are working with local campaigners in key seats throughout the country to make sure that Conservative MPs hear from people in their communities who are worried about Brexit.”