Any suggestion of Handley reapplying for the role had not been discussed.
Woods said she asked for more time to review the role. She said Cabinet had signed off on the scope of the role but by the time it was offered to Handley it had expanded.
"My advice from DIA that I received when I became minister was that it had increased quite a lot from the original job description."
Woods earlier misspoke when she said there was a confidentiality agreement with Handley over the settlement he received over the handling of the role.
A spokesman for Woods said she had received incorrect advice from the Department of Internal Affairs, and as a result, misspoke when she responded to reporters' questions on the issue earlier today.
Woods was speaking to reporters after Handley released a tranche of communications between him and former minister Clare Curran, and Prime Minister Jacinda Ardern.
He received a payment of $107,000, which will be donated, after he was offered the role by Curran in August when she was digital services minister, then the offer was withdrawn after she was sacked as minister.
A spokesman for the minister later said she had based those comments on incorrect advice from officials and after seeking more information, had receive clarification.
Curran said today it was Handley's right and his choice to release the information. "I think it's good that it's out there."
Curran said she and the Government were "working through" releasing the information themselves.
"With relation to the Government's process, it is a different process and there are other responsibilities and obligations, and one of the obligations that I had was to respect the process with regard to him."
She denied that it was embarrassing for Ardern.
"No, I think this is part of what happens when there is a controversial issue and transparency is actually occurring. I would like to make the point that the State Services Commission found that the process of the recruitment was very robust and the meeting I had with him in February did not prejudice that process."
An aide-memoire from the State Services Commission to Minister Chris Hipkins earlier this month said the evidence suggested a "suitably robust " recruitment process. It also said that viewed objectively, the meeting did not prejudice the process.