Bloomfield confirmed his team had come up with advice about tightening alert level 2 restrictions, including mask use. Robertson said any decisions on tweaks would be made tomorrow.
Bloomfield said 38 people were now in hospital including six in intensive care, with four requiring ventilation. There are now 801 cases in the outbreak, including 79 recoveries.
The 20 cases today matches the same number as that released on Saturday, and down from daily highs in the 80s a week ago, and 75 last Wednesday.
Bloomfield said the ongoing drop in numbers showed level 4 lockdown measures were working but the highly contagious nature of the virus meant it was important to remain vigilant.
"We are heading in the right direction," said Robertson. "But the job is not over yet. This is why we want to see testing numbers in Auckland increase."
Border screening for essential travellers, workers
Bloomfield said the Government wanted to make sure there was no "leakage" across the Auckland boundary, by screening people who were crossing in and out of the region.
Bloomfield said rather than testing each person this would likely be screening for symptoms, "just as we do with people at the [international] border, at ports and airports".
Discussions were happening with the transport industry on screening at Auckland's boundary, Robertson said. A number of businesses were supportive, he said.
"We're not suggesting that you'd be stopping at the boundary to be tested, because that would create a level of congestion that would not be helpful," he said.
"It's not so much testing at the border, but surveillance testing of people crossing the border," Bloomfield said, adding it wouldn't be "a forever thing".
Robertson reiterated the need to stay vigilant despite the drop in daily cases. He said Victoria relaxed restrictions and was now reporting hundreds of cases per day.
He urged people who were a close contact to get tested, along with anyone who had cold or flu symptoms. "With Delta, there is no room for complacency. Let's double down, let's get the job done. We have got to do this once and we have got to do this right," said Robertson.
On the death of a North Shore woman, aged 91 - the first death in this outbreak - Bloomfield endorsed the woman's family's message for others to take the virus seriously and follow public health advice.
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Bloomfield said ongoing investigations showed the total number of unlinked cases had fallen from 58 last Sunday to 30 today.
Analysis of yesterday's cases show 75 per cent were contacts of cases. Six people were potentially infectious in the community.
There were several thousand community swabs taken yesterday, Bloomfield said. Auckland Regional Public Health Service analysis of testing showed more than 20 per cent of the Auckland population had been tested.
Robertson said "a significant amount of effort" had been put into supporting people in Auckland's largest sub-cluster, the church cluster. He said organisations like South Seas were all actively engaged with that cluster of people.
'Lockdown is doing its job'
The downward trend in cases has prompted modeller Shaun Hendy to say lockdown is "doing its job against Delta", and the possibility of Auckland moving out of alert level 4 this month is "certainly looking more possible".
But experts fear there could still be a long tail, and the current outbreak could continue to produce more cases for weeks.
Meanwhile, August 20 was the last time any Wellington case or contact was active in the community, Bloomfield said.
Robertson gives update on terror attack
Robertson also gave an update on the supermarket terror attack.
No Covid patients had to be moved out of ICU for the terrorist attack victims, Bloomfield said. All three major hospitals in the Auckland region have capacity.
The terrorist, who travelled to New Zealand in 2011 and later received refugee status, came to attention of authorities in 2016 due to his online activity, Robertson said. "We are not looking for anybody else in this situation," he reiterated.
"At every opportunity, we have been looking at ways to deport this individual," Robertson said. The PM raised the issue in the first meeting about him in May 2018, Robertson said.
Robertson said the terrorist was in prison for a large time, and the deportation process stopped because of that. "We have at every turn... gone to every part of the law," he said.
Robertson said he understood some documents the terrorist used to get refugee status had been fabricated, and on that basis Immigration New Zealand had looked into his status.
He did not have details on the alleged fraudulent documents used by the terrorist but a "drawn-out process" then followed.
Robertson said a person could not be detained under the Immigration Act for anything other than deportation. "We are looking at the full sweep of the Immigration Act," Robertson said.
The terrorist's deportation was delayed because he faced criminal charges that "were still live". While the terrorist was in prison, he was largely there on remand, said Robertson, while the legal process was happening.
Robertson said there were attempts throughout the process for mental health issues to be dealt with, and no attempts were able to change the state of mind of an individual with his ideology.
Robertson said at every turn the Government tried to find a remedy to the current legislation.
It was a court order that put him back out into the community, Robertson said, not the Government.
Robertson said he was not aware of any contact between the Government and the man's family, apart from the courts who let his family know about the lifting of suppression orders.
Cabinet will consider tomorrow the possibility of a wider investigation following the man's death. There were a range of reviews that are possible, Robertson said.
Robertson said people outside of INZ recognised possible fraudulent papers used by the terrorist, but he said the current refugee application process was a rigorous system.
Robertson began today's briefing, wishing all dads a Happy Father's Day, as well as birthday wishes for his mother and fellow minister Chris Hipkins. He also paid tribute to essential workers, including police, ambulance officers and supermarket workers associated with the terror attack, as well as those helping the vulnerable in the Covid outbreak.
Covid case death
Today's announcement of latest Covid cases follows the death of a woman, aged in her 90s, at North Shore Hospital. She is the first person in New Zealand to die from the Delta variant of the virus.
"Every death is a reminder of the damage Covid-19 can cause when it gets into our community," Prime Minister Jacinda Ardern said yesterday.
The woman had underlying health conditions, which meant it was not clinically appropriate for her to receive ventilator or ICU care.
Her family said they were devastated and shocked by the loss of their loved one.
The Ministry of Health said one of the affected family members was urging everyone to take Covid-19 seriously and follow public health advice. His message for New Zealanders is simple: "This is real."
Northland was the latest region to move to alert level 3 on Friday, joining the rest of the country.
Auckland remains under lockdown until a review will be made by Cabinet on Tuesday September 14.
