New Zealand officials were scrambling to trace the source of a COVID-19 outbreak, reporting 13 new cases on Thursday, as long queues of people formed to escape a renewed lockdown in the country's biggest city Auckland and for testing.

The discovery of four infected family members in Auckland two days ago shocked a country that had not recorded any COVID-19 cases for more than three months, raising criticism of the government's handling of the health crisis.

New Zealand announced on Thursday that there were 13 new cases in the community, bringing the total number of active cases to 36.

Prime Minister Jacinda Ardern has swiftly reimposed tight movement restrictions in Auckland and social distancing measures across the rest of the country, echoing her early response to the pandemic, which was praised for its apparent effectiveness.

Other parts of the country were placed back into slightly looser level 2 restrictions. The restrictions will initially remain in place until Friday, when Ardern will announce the next steps.

The dissolution of parliament, the precursor to a general election that will soon take place, has also been deferred until Monday. This has raised questions over whether the government will postpone the election. While no decision on the matter has been made, Ardern said she will announce one before Monday.

With most parties having suspended campaigning after emergency levels were moved up, the major opposition party National has called on the prime minister to move the September election to November, citing unfairness for them because of the inability to campaign. 

Meanwhile, doubts over the origin of the latest cases have also raised some doubts about Ardern's strategy. Officials reported on Thursday that three more people at a refrigerator storage facility, where one of the family members worked, had tested positive.

Director-General of Health Ashley Bloomfield said health officials were getting "closer every hour" to finding the patient zero of the outbreak.

"We are working flat out to do the contact-tracing we need to do and trace back to find out what the source of this infection is," Bloomfield told Newstalk ZB radio.

Bloomfield raised the possibility on Wednesday that the virus had arrived in New Zealand via freight, given one of the infected family members works in a cool store that takes imported frozen goods from overseas.

On Thursday, he said that was considered "a low possibility," but did not detail other potential sources.

Some prominent local health experts suggested it was more likely the virus had been quietly spreading in Auckland for weeks, infecting potentially dozens of people.

Residents of Auckland, home to around 1.7 million people, were given just hours to prepare for the return to level 3 restrictions on Wednesday, requiring people to stay at home unless for essential trips.

People in the city reported waiting hours to be tested for the virus, according to local media, and there was a rush to supermarkets to stock up on essentials.

(With input from agencies)