Appearing to set out his stall as a future leader of the Conservative Party, the former Brexit secretary outlined his hopes for a "fairer deal for working Britain" and how the party could win back support from those who are less well off.
Mr Raab, who quit the Cabinet in November over Mrs May's Brexit deal, has been tipped as a potential successor to the PM, who has already said she will stand down as leader when an EU withdrawal agreement is ratified.
In a wide-ranging interview with The Sunday Times Magazine, Mr Raab set out the direction he would like to see the Tories take, including focusing on tax cuts for low and middle income families.
"I think the basic rate, taking a penny off that, would be talking to the people who need to know we are on their side," he said.
He also said the party should be at the forefront of injecting "lost" optimism about Britain and its future back into the political frame, as well as focusing on a "positive" message for hard-working Britons.
He added: "I think the Conservatives and our record on free enterprise and jobs is great, but we have got to be talking about and to the worker who hasn't had a pay rise in several years, the consumer who feels ripped off by sharp corporate practice, the young kid from the rough background who wants their shot, who wants to make the best of their potential, and I think if we can have a really positive, compelling message in all of those areas, we will unite the aspirational working and middle class in this country, which is how Tories win elections."
He also talked of hopes for all fathers to have the right to two weeks' paternity leave at 90% full pay and a change in the law to ensure new or expectant mothers cannot be made redundant during pregnancy or maternity leave.
Mr Raab has already gained the support of fellow former Brexit secretary David Davis, who said he would back the 45-year-old to become the next Conservative leader.
Writing in the Daily Mail, Mr Davis said Mr Raab was the "standout candidate" and the "best-placed Brexit candidate to win the necessary support among MPs and party members and, above all, broaden our appeal to voters".
"I have known and worked with Dominic over the last 13 years so I know he has the vision and personal attributes required to lead us at these crossroads in our history," he said.
In the wake of the local election results, former Conservative Party leader Iain Duncan Smith said the 1922 Committee of backbench Tory MPs should urgently meet again to decide on Mrs May's fate.
"We have to make a change... The message was loud and clear that, since March 29, people have decided they are absolutely furious with the political class," he told LBC.
"The committee has to sit again now, urgently, and decide that either the Prime Minister sets the immediate date for departure or, I'm afraid, they must do it for her."
The threat of an imminent challenge to Mrs May's position as Conservative leader was lifted last month when the 1922 Committee's executive rejected calls to change party rules which protect her from a no-confidence vote until December.