Sacked minister Alun Davies has written to five AMs to apologise for trying to discover their farm subsidy payments.
It comes after First Minister Carwyn Jones said it would help if Mr Davies apologised to them.
Mr Davies lost his environment post on Tuesday for putting pressure on officials to find him details of the subsidies, which were private.
Mr Jones said later he was happy to consider making changes to the way the rules for ministers are policed.
Complaints from civil servants emerged just days after the Blaenau Gwent AM was reprimanded over another matter.
In an interview with BBC Wales on Wednesday, First Minister Carwyn Jones said he had warned Mr Davies that he was "on the edge of a cliff" for a row over an Ebbw Vale racetrack last week.
Opposition parties had said he should have been sacked for lobbying Natural Resources Wales over the £280m Circuit of Wales track in his constituency.
The union representing senior civil servants has described Mr Davies' behaviour in relation to the finances request "reprehensible".
It emerged he had repeatedly asked civil servants for details of opposition AMs' subsidies, despite being told the information was private and not publically available.
Mr Jones also said he was left with "no choice" but to sack the environment minister.
He said while he had apologised to the politicians affected, it would "certainly help his position" if Mr Davies did too.
He added that it was difficult to see a way back for the former minister as his actions raised very serious issues.
"The only conclusion you can draw, and the only conclusion that any reasonable person can draw is that he wanted to use that information against the named members," said Mr Jones. "I drew that conclusion, the public are going to draw that conclusion and on that basis I'm afraid he could not remain in the government.
"I am very annoyed. If you offer someone what is in effect a second chance, I would not expect them to throw it out of the window within hours.
"It was made clear to him last week that he was standing on the edge of a cliff after what happened with the other issue, and then the following day this happened. There was no choice I'm afraid. He had to go."
Mr Jones said he could not defend or explain Mr Davies' actions, and the former minister had also been unable to explain it to the first minister when questioned.
"One of the things I could have chosen to do was to keep this quiet, but that would have been the wrong decision," he said.
"I wanted to make sure people understood what had happened, understood that where this sort of thing does happen it may be acceptable elsewhere but it is not acceptable in Wales."
However, he said while Mr Davies had broken the ministerial code in his role as a minister, he had "done nothing" as an assembly member.
"This has been referred to the standards commissioner and it's in their hands," he added.
Responding in the Senedd to opposition calls for an independent adjudicator of the ministerial code, Mr Jones said he had an "open mind" on making changes.
But he said he was uncomfortable with the term adjudicator, as that would suggest the final decision on a minister's fate would not rest with the first minister.
All three opposition parties said Mr Davies's recent breaches of the code demonstrated the need for the system to be reformed, and for a complaint made about ministerial behaviour to automatically receive independent scrutiny.
But Mr Jones said he would not have been able to deal with Mr Davies as quickly as he had if there had been an automatic referral to an independent figure.
The opposition motion to create an independent adjudicator was defeated by 25 votes to 24.