Dumped minister Jamie Briggs is expected to recontest his seat in South Australia in spite of continued pressure, including a further public reprimand from Prime Minister Malcolm Turnbull.

Multiple South Australian Liberal sources have said that despite speculation Mr Briggs will quit politics, there are firm indications he wants to fight the next election.

Meanwhile, Mr Turnbull ruled out launching an investigation into who leaked a photo of the woman at the centre of the Briggs affair.

Labor is expected to step up its pressure over the investigation issue, with Opposition Leader Bill Shorten preparing to write directly to Mr Turnbull demanding a probe.

Mr Turnbull reiterated condemnation of the leak – and singled out Mr Briggs' forwarding of the picture to colleagues – but said leak inquiries typically failed to find the culprit.

"The sharing of that photograph was very unfortunate. It was not the right thing to do, clearly I think everyone recognises that, including Mr Briggs," he said.

"But . . . these leaks inquiries, they tend to come up with very little. I think we know the photograph was taken on Mr Briggs' phone and he shared it. He shouldn't have done it, quite clearly."

Mr Briggs' spokeswoman declined to say whether he would recontest the seat at the next election, referring back to his original resignation statement in which he said: "I will be working hard in my electorate to ensure the re-election of the government."

She also declined to address questions as to whether Mr Briggs was upset by the leak of the photo, whether he knew who had leaked it, or whether he had taken the issue up with any suspects.

Mr Briggs has admitted forwarding the image – which appeared aimed at discrediting the woman who later reported that Mr Briggs had behaved inappropriately towards her – to "several" other people, but has denied leaking it himself.

Mr Briggs resigned after the woman claimed he had kissed her neck and said she had "piercing eyes" during a drinking session in a Hong Kong bar.

His likely main rival in his South Australian seat of Mayo, Rebekha Sharkie​ of the Nick Xenophon Team, said voters would "judge for themselves" whether the dumped minister's personal conduct was grounds for him to lose his seat.

Ms Sharkie, who, ironically, used to work for Mr Briggs, said she would run on the issues such as infrastructure and jobs and vowed not to raise Mr Briggs' conduct in the campaign.

She was quoted in a newspaper article last week accusing him of sexist behaviour when she worked for him but on Tuesday protested that her comments were "not construed how I meant them to be".

Labor is yet to choose a candidate for the seat, which Mr Briggs holds with a 12.5 per cent margin.

Sean Edwards, a moderate South Australian Liberal senator and former state president of the party, said he didn't know whether Mr Briggs would run again.

"I haven't spoken to Jamie. But I was in the electorate of Mayo on New Year's Day and Mr Briggs enjoys a great deal of support among the main business and community people I spoke to," he said.

Labor also hit out at the government for failing to appoint a new sex discrimination commissioner, linking the continued delay to Mr Turnbull's "lack of action" on the Briggs affair and the controversy over Immigration Minister Peter Dutton's description of a female journalist as a "mad f---ing witch".

Former commissioner Elizabeth Broderick's term expired on September 3.

- with Fergus Hunter