Mr Turnbull became the fourth prime minister to be dumped by his or her own party since 2010 in response to poor opinion polling.
The by-election date will be set by the Speaker, Tony Smith, after Mr Turnbull formally resigns.
"I want to use his experience", Mr Morrison said on Monday. "I can not remember a single positive policy initiative that Abbott has championed and then implemented", Mr Rudd said.
Scott Morrison was sworn in as the country's 30th leader late Friday after a bitter Liberal Party revolt against moderate Turnbull driven by a hardline conservative faction.
Scott Morrison has unveiled his new ministry.
If the Liberals lose Wentworth, the Coalition will hold 74 lower house seats and will rely on Nationals MP Kevin Hogan and Bob Katter as deciding votes.
"I don't think any of us here needed an opinion poll to tell us how angry, confused and annoyed Australians are with the Government", Mr Shorten told reporters yesterday.
Mr Morrison, whom Mr Turnbull described as a loyal treasurer, excluded Mr Abbott from his first cabinet but has offered him a newly-created role of special envoy for indigenous affairs.
"As a former road transport operator, I understand industry issues and know how important a good, reliable road and transport system is to communities big and small right across Australia", he said.
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Tony Abbot, meantime, has backed his sister Christine Forster, who is being encouraged to seek Liberal endorsement.
"I've had a great team in federal parliament and a great team here at Wentworth", Turnbull said last night.
Mr Bragg resigned from his job at the Business Council of Australia on Tuesday, telling colleagues he wanted to focus on winning preselection to contest a byelection, expected in October.
Broadcaster Ray Hadley - under fire for encouraging MPs to abandon Turnbull in favour of conservative Peter Dutton - told Abbott the role was intended as an olive branch and sign of respect.
Dave Sharma, Australia's former ambassador to Israel, has also entered the race.
Ms Forster has been a Liberal councillor for the City of Sydney for the past six years and was a leading figure in last year's successful same-sex marriage campaign.
"I always suspected that the instant he didn't have the top job he'd want to go", he told radio station 6PR.
Mr Turnbull will resign on Friday, triggering a by-election in his Sydney electorate of Wentworth.
Dutton, who tried to unseat Turnbull, was re-appointed as home affairs minister, a post he previously held during the Turnbull administration, but without the portfolios of Immigration, Citizenship and Multicultural Affairs, which were taken over by David Coleman.