Prime Minister Jacinda Ardern says she wants to solve the issues which let two COVID-19 positive women leave isolation before being tested, rather than punish people.
In a press conference on Thursday, Ardern acknowledged the case was a "failure of the system" but said she is focused on fixing the problems, not finding the people responsible.
"I'm not interested in going down to individuals and finding out exactly who [was involved]. We do need to know what happened so we can fix it, it's not a witch hunt," she said.
Ardern admits confidence has faltered since the new cases and said she is focused on restoring it.
"We have had a setback in people's confidence, it's our job to restore it."
The Assistant Chief of Defence, Air Commodore Digby Webb, will take over the management of quarantine facilities starting with "an audit function of the end-to-end process of quarantine facilities - including the health checks".
The two women, who arrived in Auckland from the United Kingdom, applied for compassionate exemption to attend a family member's funeral in Wellington.
They travelled from Auckland to Wellington in a private vehicle and had contact with two friends before they tested positive on Tuesday.
New Zealand was declared COVID-19 free on Monday June 8, that ended on Tuesday when the two women tested positive. Another person tested positive on Thursday.