He said told the Evening Standard: "We on the centre-Right side of the argument have to have just as inspiring a vision - a more inspiring vision - of how you build not just a strong economy but a strong society and a better life".
The Standard is now edited by George Osborne, who served as Chancellor under Mr Cameron and has become an outspoken critic of the Tory government under Mrs May.
Asked why young people were associating idealism with Corbyn's Labour and not the Conservatives, he said: "I think we need to push it more".
Speaking to the newspaper after chairing a board meeting of patrons of the National Citizen Service (NCS) scheme he founded, Mr Cameron said: "It is very important that the Conservative Party doesn't slip backwards".
He added: "The reason I wanted to lead the Conservative Party back in 2005 was that I wanted us to be more than 'the economics party", more than just free marketeers with the rough edges knocked off.
"Modernisation isn't an event. A political party should be asking itself all the time, 'Am I properly in touch with and reflecting the society and the country?'".
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The former Tory leader insisted the party must continue to be the "open, tolerant and liberal" party he rebuilt in 2005.
The former prime minister suggested the Tories' failure to make an economic argument during the election meant voters had "forgotten just how risky this full-on programme of nationalisation, state control and rampantly high taxes can be".
"You don't win the argument in favour of free enterprise, free markets, choice and liberal democracy and then pack up and go home", Cameron said.
Mr Cameron also joked that he would like to strap some of his former Cabinet ministers on a raft and send them down a "very, very unsafe river". "You have to win the argument in every generation".
And in a blast at warring Cabinet ministers, he joked about strapping them to a raft on a "very, very unsafe river" as an exercise for youngsters on the National Citizen Service he created as PM. I think that's still very true today'. "They are in the grip of a minority who seek hard Brexit at any cost", it said.
The former prime minister said it had been "depressing" to see his party lose seats in London and other metropolitan areas in the recent general election.
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