Prime Minister Jacinda Ardern is asking residents in seven Auckland suburbs to be "especially vigilant" as the region enters another week in lockdown.
Seven Auckland suburbs will be the focus of testing and contact tracing this week as health officials try to crack the mystery Covid cases.
"Please stick to your bubble. Stay home as much as possible. Try to reduce risk by having just one person go to the shop or supermarket," Ardern said.
Health officials are on the hunt for the virus in Mt Eden, Massey, Māngere, Favona, Papatoetoe, Ōtara and Manurewa.
That was where they had the biggest concerns about the 34 cases with no known links to the current clusters.
Ardern asked residents in Mt Eden, Massey, Mangere, Favona, Papatoetoe, Otara and Manurewa to be especially vigilant.
Auckland will remain at level 4 for another week, until 11.59pm on September 21.
Cabinet has made an in-principle decision that Auckland will move to level 3 at that time.
"The next week will be critical in providing the additional reassurance we need," Ardern said.
Auckland councillor for Manukau, Alf Filipaina, has four of the suburbs in his patch.
People were nervous, he said.
"They're feeling anxious because we've been in lockdown now since the 17th of August," he said.
People were keen to do what it took to get out of level 4, he said.
He supported the increased testing - and wanted more vaccinations too.
Health authorities had already been surveillance testing - looking for the virus in asymptomatic people rather than waiting for it to turn up when people get sick.
Last week they were at essential workplaces, including Countdown's distribution centre and its two big warehouses for online deliveries.
Auckland University public health lecturer Colin Tukuitonga said the testing should also include supermarket customers in the areas of interest because that is one place where a lot of people go at alert level 4.
There was also continual testing of border and health staff as well as new testing for those crossing the city's boundaries for work.
Director-general of health Dr Ashley Bloomfield said there was no evidence of widespread community transmission in Auckland.
But Dr Tukuitonga said the city was now facing a "classic long tail" of Covid, where case numbers could take a long time to fall away, and jump around.
"The worst case scenario is that we are missing a cluster or sub cluster somewhere," he said.
He worried about the cases that turned up out of nowhere to Middlemore Hospital over the past week - although some of them had now been linked to existing clusters.
All the mystery cases could be solved in just a few days if all went well, and more aggressive contact tracing was needed to really hone in on them, he said.
Auckland epidemiologist Rod Jackson said the Delta variant was extra challenging for the tracers because people became infectious more quickly.
"They're racing against time to find your contacts before they become infectious because as soon as they become infectious then it can go crazy," he said.
As well as focusing on this outbreak, the city - and the rest of the country - must prepare for the inevitable next one, he said.
That meant vaccinations.
There was a big push in Auckland, with nearly 74,000 people vaccinated from Friday to Sunday, thousands of them via drive-throughs.
But professor Jackson wanted that upped even more - with round the clock vaccination centres and weekly pop ups in all large workplaces.
Councillor Alf Filipaina agreed there must be options to suit everyone, including going to people's homes or setting up neighbourhood centres.
- Additional reporting with RNZ