The Conservative leadership outsider Rory Stewart, who has been unexpectedly catapulted into the next round of the contest, vowed that he would set up “an alternative parliament” if the frontrunner, Boris Johnson, suspends parliament to pursue a no-deal Brexit.

Stewart, who is understood to have drafted a concession speech before unexpectedly scraping through the first round on Thursday, earned one vote fewer than Matt Hancock and four less than Sajid Javid.

His campaign said it showed momentum was behind him due to his public appeal, despite being the lowest-scoring of the seven candidates to go through to the second round.

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In a plea to supporters of Jeremy Hunt and Michael Gove, Stewart’s backer, the justice secretary, David Gauke, suggested they would not have a chance of beating Johnson in the final round. “A safe pair of hands is not going to beat Boris Johnson,” he said.

Stewart said he was “absolutely over the moon” to have secured enough votes to go through and said he would be reaching out to “every single colleague” to persuade them to back him.

He said if Johnson attempted to prorogue parliament, he and other MPs would be prepared to “bring him down”. “If he were to try, I and every other member of parliament, will sit across the road in Methodist Central Hall and we will hold our own session of parliament,” he said.

Campaign sources for Hunt, Gove and Javid conceded there was now a race for second place behind Johnson, who won 114 votes, well ahead of his rivals. However, in the immediate aftermath, no campaign appeared willing to try to thrash out a consensus behind one candidate.

Play Video Rory Stewart launches campaign for Conservative leadership – watch live

Gauke said colleagues who wanted to properly challenge Johnson should back Stewart because of his engagement with the public, demonstrated in a campaign that has been dominated by shaky handheld videos of Stewart meeting members of the public in different cities.

“Here, there is a comparison to be made with Boris – Rory has that ability to connect with the public, he has got that authenticity that the public respond to really well, and he is the one capable of causing surprises,” said Gauke.

“It’s very clearly going to be Boris versus someone else. Boris is the overwhelming favourite. Rory is an unconventional candidate and he is the only one who could pull off a surprise. He is the only other candidate that people are talking about in the supermarket queues.”

Gauke said he believed there was still “quite a lot of hidden support there for Rory”, who won votes from 19 Tory MPs – 12 more than had publicly declared support for him.

Stewart’s camp has also been buoyed by a poll of party members from ConservativeHome that put him as second favourite, albeit more than 40 points behind Johnson.

However, party critics have pointed out he has failed to gain any momentum from the bloc of party moderates, with most of his backing coming from personal friendships such as those with Gauke and the Tory grandee Nicholas Soames, as well as the handful of Tory MPs who have consistently opposed a hard Brexit.

“He may be hoping to pick up Matt [Hancock’s] support if he drops out of the race, but the reality is it would be humiliating to back the only candidate who got fewer votes than you,” one MP said.

“He is getting better polling because people are starting to know who he is. That does not mean MPs think he has any chance at all. And most MPs will not be around until the next round of voting, most will be trying to switch off their phones over the weekend and get away from all the horse trading.”

But Stewart’s backers believe he can capitalise on the planned TV debates, one of which is being hosted by Channel 4 on Sunday. Johnson’s team has not confirmed whether he will take part, though the pressure on him from his rivals is mounting.

Amber Rudd, Hunt’s most high-profile backer, said it was essential that candidates committed to taking part in public debate. “I want them all to do the TV debates. Jeremy has said he wants to participate, he’s still in negotiations, but the key thing is that all the candidates get out to TV debates,” she said.

“I keep on saying the fact is, the Conservative party needs to remember that we’re not just choosing a leader, we’re choosing a prime minister, and the public need to see them. I think they’ve got a duty to do it – to be publicly interrogated.”

It is unclear, however, whether Hunt will take part in a debate if Johnson does not attend. Sources close to the foreign secretary have said he believes all candidates should take part – but would not commit to participating if his closest rival is empty-chaired.

In a video thanking supporters, Gove said there should be a “proper debate about ideas” in the party, suggesting he too would be piling pressure on Johnson to take part in the debates.