Further results of Covid-19 testing in the community due back today will provide a clearer picture of whether Auckland can be moved out of alert level 3 at midnight on Wednesday.
Cabinet will meet again today as it continues to review the alert levels every 24 hours.
Yesterday, there were no confirmed cases of Covid-19 in the community, outside of the three already reported.
Testing as of yesterday afternoon found that two of the 33 close contacts at the daughter's school had come back negative, with the rest pending.
Seven of the nine co-workers of the mother have also come back negative.
An additional nine close contacts outside of the family and school have also tested negative.
Prime Minister Jacinda Ardern said the test results so far were heartening but potential chains of transmission in the community needed to be ruled out.
"We do want to give it a bit more time before we get too far ahead of ourselves, I can't help but reflect on previous experience as we all do," she said.
"Very early on for instance in August, when we had our first positive cases reported, we had springing up around it symptomatic positives as well."
Ardern said the testing at Papatoetoe High School, the workplace, and extra community surveillance would all add reassurance.
"We are looking proactively at whether or not we need to, as an extra precaution, go through and undertake another out-of-cycle range of test for all our airside staff as well," she said.
She said a picture was being built as they went.
"But we will keep up in earnest those 24-hour reviews," she said.
Read more on the latest Covid-19 cases:
- Follow RNZ's live blog for the latest Covid-19 updates
- View updated locations of interest here
- Alert Level 2 and Level 3: What they mean
Meanwhile, Parliament will this afternoon pass a bill giving businesses support in the event of a longer lockdown.
The bill will be debated under urgency and pass all stages today.
Ardern said the planned framework for financial support was announced last year, to give businesses certainty about what would happen if there was another lockdown.
Support would kick in if alert levels were raised for a week or more, she said.
That includes the wage subsidy for regions at alert level 3 or higher, at the rate of $585 a week for every full-time employee and $350 per part-timer.
Businesses will need to show a 40 percent decline in revenue compared with the six weeks prior to the alert change.
Those at alert level 2 or higher can receive a one-off resurgence support payment if they experience a revenue drop of 30 percent or more over a 14-day period.
With the resurgence payment, businesses will get $1500 as well as $400 for every full-time employee (up to 50 employees.)