Ten of the new cases were either cool store staff or their family members.
Authorities on Tuesday had confirmed four cases of the virus in one Auckland household from an unknown source.
Since then, the outbreak has grown to 29 people and extended beyond Auckland.
"Two of the AmeriCold sites, Mount Wellington and Auckland Airport, have been closed, and all staff from the Airport site have now been tested", the NZ Health Ministry said on Thursday.
Of the 13 new cases, all bar one, are linked to existing cases.
There are also four new cases of employees at two Auckland businesses and eight family members of those employees, all linked to the index case.
"But we are not out of the woods yet". The schools are Glamorgan School on the North Shore and South Auckalnd's Southern Cross campus, and Taeaofou I Puaseisei Preschool.
"Those who have flu-like symptoms need to contact their GP or Healthline and get tested".
This means people living in Auckland are working from home unless they are an essential services worker. "We should expect that to be the case here". The source of their infections has not been traced.
All the 600 families and staff at Glamorgan School in Torbay have been ordered to stay in self-isolation. Arrangements are being made to establish a bespoke quarantine arrangement for their wider family and a testing station will also be up and running in Tokoroa over the weekend.
Bloomfield said that they completed a record of more than 15,000 tests Thursday and they were getting a clearer picture of the outbreak's contours.
Over the past 24 hours, some 180,000 people have downloaded and registered the app, bringing the total number of users to 1.17 million. "Tthere are 16 stations around Auckland and it is free at your local GP".
There are 30 active cases connected to the recent outbreak. "The virus is the problem, not people". Health workers - especially those involved in testing - have been subjected to abuse, he said.
Mr Seymour also said people in Auckland, mainly Epsom where he is based, were "absolutely flabbergasted" at the performance of some in Government.
"That will mean it will be very hard for some other people to get the medicines".
He pointed out that pharmacies would remain open.
There are 12 new confirmed cases and one probable case of COVID-19 to report in the community today - 13 new cases in total. That prompted Ardern to reintroduce coronavirus restrictions, including shutting off access to retirement homes, imposing the Auckland lockdown, and limiting gatherings around the country to no more than 100 people.
Later today at 5.30pm, the country will find out about any further alert level changes. This possibility has allowed the major opposition National Party to seize on potential flaws in Ardern's approach ahead of the September 19 election, which she described as a "Covid election" when launching her campaign.
Cook Islands Tourism chief executive Halatoa Fua would not be drawn on the air-bridge delay, saying Deputy Prime Minister Mark Brown and Secretary of Health Dr Josephine Aumea Herman's comments were sufficient for now.