Peter Dutton is facing the claim he intervened to help a third European au pair, in fresh evidence from the former Border Force commissioner Roman Quaedvlieg to a Senate inquiry investigating the controversy.

In a submission published on Monday, Quaedvlieg clarified earlier evidence that he fielded a phone call from Dutton’s office in June 2015 by suggesting his memory related to a new unreported case between October 2015 and 2016.

While suggestion of a third case emerged as a fresh threat, Dutton attempted to fend off a call for a no-confidence motion based on his evidence to parliament he had no “personal connection” to employers involved in the first two au pair visa cases – in Brisbane in June 2015 and Adelaide in November 2015.

Dutton’s travails were the focal point of a challenging first parliamentary day for the new prime minister, Scott Morrison, who was pressed during question time to explain why he was now the prime minister and Malcolm Turnbull was not.

With a new poll suggesting the ALP would easily win any election held today, the opposition targeted Morrison and the ministers who pledged loyalty to Turnbull in the House of Representatives before voting for Dutton in the three-way leadership ballot.

Morrison said he was in the top job because that was the will of his Liberal party colleagues. “John Howard used to say something quite simple and that is the privilege of serving as the leader of your parliamentary party is the decision of your parliamentary party,” he said.

The prime minister also continued to back Dutton in the face of the parliamentary attack.

On Monday, Dutton tabled an email from a former colleague in the Queensland police, Russell Keag, asking for assistance in June 2015, which bolstered Labor and the Greens’ case.

In the email dated 17 June 2015, Keag opens by observing it has been a “long time between calls” and states the facts of the case with a request for “advice” from Dutton.

The Greens MP Adam Bandt seized on the email, arguing it showed not only that Dutton had “a personal connection with the man he helped, contrary to what he told [parliament] in March, but that they spoke to each other too”.

“If he won’t resign, the [prime minister] must sack him,” he said.

It now appears that Peter Dutton didn't just have a personal connection with the man he helped, contrary to what he told Parl in March, but that they spoke to each other too, contradicting his 'no personal contact' claim to Parl today. If he won't resign, the PM must sack him. https://t.co/nPNx58FZaW — Adam Bandt (@AdamBandt) September 10, 2018

Labor’s immigration spokesman, Shayne Neumann, said the “explosive email” cast further doubt over Dutton’s claims he had no personal connection to the employer of the au pair.

“After reading this email, how can Scott Morrison have any faith left that Peter Dutton should remain the minister for home affairs?”

Earlier, Dutton made a statement to the lower house denying that he had misled parliament over the first two au pair cases and reiterating that he had no “personal connection or any type of relationship” with the people involved.

Dutton told parliament that, when his staff member asked if he knew Keag, his “initial response was ‘who?’”.

Despite the revelation that Dutton had worked with the employer of the au pair in the June 2015 case, the home affairs minister said a “personal connection” signified a “much closer relationship between two people than working in the same organisation two decades ago”.

“To the best of my knowledge I have not socialised with, met with or had personal contact with the man involved,” he said.

Dutton said the former colleague did not have his personal phone number or email address, and had contacted him via his publicly available email account.

In the Adelaide case – in which the Australian Football League chief executive, Gillon McLachlan, asked the AFL’s head of government relations to contact Dutton on behalf of McLachlan’s second cousin – Dutton said he “never met the visa holder or anyone else involved in the matter”.

On Wednesday, McLachlan told a Senate inquiry hearing he believed he had met Dutton on about six occasions, and recalled three specific instances of meeting him including when Dutton was sports minister.

Dutton claimed that he signed off on both cases after the immigration department presented submissions that had “a recommendation to intervene … on the front page”.

Before the tabling of the Keag email, the threat of a no-confidence motion against Dutton looked likely to subside because the Centre Alliance MP Rebekha Sharkie said she accepted his explanation.

In relation to the fresh case, Quaedvlieg repeated the central assertion that Dutton’s chief of staff, Craig Maclachlan, had contacted him to request help for “the boss’s mate in Brisbane”, adding that he found it “unusual” the chief of staff would call about a “low-level transactional issue”.

Quaedvlieg said without access to departmental records he wasn’t sure of the details of the case, but suggested it related to “a young European female travelling on a tourist visa … from a western or southern European country”.

Records of ministerial intervention show that Dutton did intervene in a case in June 2016, within the range of October 2015 and the end of 2016 nominated by Quaedvlieg as the likely period of the new case.

Quaedvlieg explained he had “made the apparently erroneous assumption” the call from Maclachlan related to June 2015 because of the “remarkably coincidental circumstances”.

After he was contradicted by Dutton, who noted Maclachlan was not his chief of staff at the time, Quaedvlieg suggested that departmental records “need to be examined to correlate it to a different Brisbane visa case”.

The Senate inquiry examining the au pair cases has asked to extend its inquiry to next Wednesday.