Theresa May must remake the case for conservatism to take on Labour and prove that her party cares about “left behind” parts of the country, her de facto deputy prime minister has said.

David Lidington, who oversees both the domestic agenda and the Brexit process, said the Conservative party had not yet done enough to reach voters who felt their local areas had been neglected by the government.

The Cabinet Office minister said May should go back to “first principles” as he suggested she would serve a full parliamentary term despite her leadership coming under sustained attack from within the party.

He indicated May was prepared to make more concessions on her Brexit plans in order to get a deal, in spite of fierce opposition from Brexiter MPs and the Tory grassroots.

“Yes I do think we’re going to get through to 2022,” he said in an interview. “I think people underestimate the degree to which most Conservative MPs, and certainly most Conservative members in the country, want to get behind the government and the PM and get things done.

“When you get to the next election, of course the public will have a memory of how [Brexit] was handled, but actually they, in my experience at elections, tend to ask: ‘What are you going to do next for me?’ What we offer on domestic policy is very important.”

He added: “These are things that she [May] cares deeply about. We are all in government having to spend a lot of time and energy on EU exit stuff at the moment … But I think she will also want to focus on and get results in some of these key areas of domestic policy.”

Earlier in the week Lidington raised speculation that May knew her leadership days were numbered when, asked whether she should take the Tories into the next election, he said she would “decide in due course what she wants to do”.

He told the Guardian: “If I were to say no, which I’m certainly not going to say, that then becomes dagger-wielding. If I say yes, that’s turned into a great plot of some kind. So as long as she’s got the energy to continue, she’ll have my support to do it.”

His intervention follows what was widely regarded as Jeremy Corbyn’s most successful Labour conference yet. Prominent Tory ministers and backbench MPs have been left fearful that Labour’s policies have exposed a lack of substance in their party beyond infighting over Brexit.

As May heads this weekend to what many predict will be her final party conference as leader, Lidington stressed the importance of making the Tory case for government.

“I think we can never do enough, let me put it that way. There’s always more that can be done and it was clear at the last election, as the results showed, that we were not getting that message across with sufficient persuasive power, and we were not reaching those people whose votes were perhaps affected by those issues,” he said.

“As a party, I think we have to make again, from first principles and telling stories to illustrate it, the reasons why we believe that a free-enterprise economy is the best way to deliver both high living standards in the face of global competition and digital technology, and at the same time enable individual men and women to have greater freedom for themselves and their family. That they are better doing that at company level, at the individual, family level, than if this is done here in Whitehall.”

He warned the Tories against relying on outdated tactics to attack Labour. “What is true is that some of the arguments that we’ve used about Labour in the past won’t work now,” he said. “With every year that passes there’s less and less point in referring to the 70s. You can’t just republish an Adam Smith Institute pamphlet from 1975 and think that’s going to do it.

“You’ve got to take people through the argument from basics. You’ve got to take account of the reality of [the] gig economy, of the fact that young people will expect to change not just jobs but careers several times in their lifetimes.”

David Lidington: ‘What we offer on domestic policy is very important.’ Photograph: David Levene/The Guardian

Lidington, a former Europe minister under David Cameron, is part of May’s inner circle on Brexit, one of just a handful of ministers trusted to meet Brussels chiefs and oversee Brexit plans on her behalf.

He remains confident of getting a deal, although government insiders have suggested this could ultimately be dependent on Brussels compromising on its proposals for a backstop for the Irish border. He said conversations had “intensified” since the disastrous Salzburg summit, where EU leaders rejected key planks of May’s Chequers plan.

This week the foreign secretary, Jeremy Hunt, said Chequers was only the “basis for agreement”, raising the prospect of further concessions from May despite deep resistance from Tory Brexiters. “The prime minister has said repeatedly that nobody is going to get all that they want, there’s going to have to be give and take on both sides – there always is in a negotiation,” Lidington said.

However, he was clear that the UK would not break its word by under-cutting the EU on social, environmental and employment standards, which some Brexiters and libertarian MPs believe could boost the economy post-Brexit.

“It’s not just about the standards of production, it’s about the environment in which that production takes place and whether you unfairly undercut the costs, becoming a sort of Shanghai on the Thames or Shenzhen on the Thames. But that is not where the prime minister is. The Chequers package represented a strategic choice for continued close alignment with the European model.”

Lidington claimed that getting the final deal through the House of Commons was not something that kept him up at night, suggesting that once a final deal was agreed it would focus the minds of MPs on all sides.

He hinted that the political declaration on the future trading relationship with the EU that accompanies a withdrawal agreement could be deliberately vague, even though No 10 is understood to have concerns about getting it past MPs.

“You can’t have, you won’t be able to have, a document this year which is able to answer every detailed question,” he said. “But it’s got to have enough clarity so that parliamentarians both here and in Strasbourg know where it is we’re heading. It’s got to be something better than saying this is going to be the biggest, most beautiful agreement that anybody has ever reached. It’s got to be more than slogans.”

Lidington said there was no getting away from the fact that an abrupt Brexit would involve “considerable economic dislocation”, which in the long term could affect investment in some of the “left behind” parts of the country, many of them leave-voting.

Many of May’s MPs feel she has not done enough to offer solutions to those areas, but Lidington said: “We don’t think that can be done by passing an act of parliament or waving a magic wand. It is more complicated. I don’t think it’s being honest to people if you’re saying this can be done overnight, click the fingers. We’re talking about very deep-rooted problems here.”

But he conceded: “It’s not enough to say that average prosperity has increased. Actually, you can’t be comfortable with a state of affairs in which there are whole communities and sectors of society that feel left out.”

