Former Brexit secretary Dominic Raab has formally entered the Tory leadership race with a call for a “new direction”.

Raab said he would prefer to leave the EU with a deal but said the UK must “calmly demonstrate unflinching resolve to leave in October – at the latest”.

The MP for Esher and Walton, who resigned over May’s withdrawal agreement, told the Mail on Sunday: “The country now feels stuck in the mud, humiliated by Brussels and incapable of finding a way forward. The prime minister has announced her resignation. It’s time for a new direction.”

Earlier, Matt Hancock, the health secretary, became the latest cabinet minister to declare he will stand for the Tory leadership after Theresa May’s resignation.

“Yes. I’m going to run to be the next prime minister,” Hancock told the BBC on Saturday morning. He said he would be “the servant of parliament” in delivering a Brexit deal – a conundrum that destroyed May’s leadership.

Hancock, 40, joins a long list of prominent Conservatives vying for the leadership of the party – and, by default, the country – including Boris Johnson, Rory Stewart and Andrea Leadsom.

The international trade secretary, Liam Fox, has refused to rule himself out as a candidate.

Asked if he would stand, Fox told the BBC: “Well, I don’t think it’s likely to happen, but, as you say, it’s an unusual contest.”

Priti Patel, the former international development secretary, told the Daily Telegraph on Saturday that she was considering standing for the leadership.

Despite May’s troubles in winning over enough MPs for her Brexit deal, Hancock told Radio 4’s Today programme that a majority of the Commons wanted to leave the EU, but that there had been “disagreements on how”.

He rejected suggestions that a general election should be held, saying it would be “a disaster for the country” and would risk “Corbyn by Christmas”.

Speaking on the same programme, Amber Rudd, the former home secretary, who will not be standing for the top job, said she would work with any prospective leader who could find the centre ground that would unify parliament behind Brexit, but refused to name anyone.

She also dodged questions about Johnson. Asked whether she thought him an honest politician, she replied: “There are some good candidates out there.”

“I’m not going to start maligning any of my colleagues,” Rudd said. “We’ve got a leadership race on. This is a really serious moment for our country.”

Nor would Rudd be drawn into commenting on May’s record as home secretary, even when goaded with the mention of the controversial hostile environment policies that led to deportations of people to the Caribbean who had lived in the UK for decades. Instead, she preferred to highlight May’s impact on raising the ambitions of young women in politics.

She said: “I will leave other people to make their points. My point on the day after she stepped down is to highlight the really powerful impact she had.”

Earlier, Stewart said he could not serve in a government led by Johnson, who has pushed for a no-deal Brexit. Johnson has said he would take the country out of the European Union by October, “deal or no deal”.

Stewart, the international development secretary, told Today: “I could not serve in a government whose policy was to push this country into a no-deal Brexit; I could not serve with Boris Johnson.”

With Nigel Farage’s Brexit party expected to perform strongly when the results of the European elections are revealed on Sunday night, prime ministerial candidates will come under intense pressure to take a tough stance on Brexit.

In a statement outside 10 Downing Street on Friday morning, May said: “It will always remain a matter of deep regret to me that I have not been able to deliver Brexit.” She insisted, however, that she had pursued the right strategy until the end.

Nominations for the leadership contest will close in the week of 10 June, with MPs involved in a series of votes to whittle down candidates to a final two contenders. Tory party members will then decide who wins the runoff.