So far, five close contacts of the cases have been identified.
There is also one new case: a returnee in a managed isolation facility.
McElnay urged anyone being cared for in the community with Covid to seek medical help anytime they need it.
"People have died this week and that is tragic for their friends and family. It is a sad reminder that Covid is potentially fatal, and this is particularly true if you are unvaccinated," she said.
There are 81 people in hospital, including 11 in ICU. A total of 51 of those in hospital had not been vaccinated, 20 have had one dose of the Pfizer jab, while one has had two doses, but the second dose had been given to them less than 14 days ago.
A further five fully vaccinated people who have been diagnosed with Covid are also in hospital.
ICU occupancy is around 70 per cent, and ventilator capacity is at 15 per cent.
"We can manage, but it is important that we limit the loads on our hospitals."
One of the six new cases in Stratford is in hospital and in a stable condition, and the other five were isolating at home, McElnay said.
Covid has also been detected in the wastewater system in Taupō.
There continues to be high demand on testing; 30,650 tests were taken across the country yesterday, including 11,683 in Auckland.
There are 18 community testing centres open across Auckland today.
There are almost 3000 people isolating at home. That includes 1382 Covid-positive cases across 920 homes.
In Taranaki, 86 per cent of people have had at least one dose of the vaccination, and 73 per cent have had two doses.
Among Māori in Taranaki, 73 per cent have had at least one dose, and 54 per cent have had both doses.
In the Waikato, four of today's 15 cases have been linked back to known cases, while the other cases are unlinked.
McElnay said health professionals were being put under "considerable pressure" to give people exemptions from the vaccine. However, the application and process to be exempted would only be approved by a single national body set up for the task, McElnay said.
Deputy Prime Minister Grant Robertson said it was an important milestone yesterday for New Zealand to reach 90 per cent of people having had one dose and 80 per cent a second dose.
He said many DHBs across the country were close to hitting the 90 per cent single dose mark. He urged those unvaccinated to go out and get the jab this weekend and help their region hit the milestone.
About 20 per cent of those infected with Covid in the Delta outbreak have been aged under 12, and included babies.
That's why the Government had introduced mandatory vaccination for education staff.
That was the same with health workers, Robertson said. Having them vaccinated gave those ill in hospital and other health care environments an extra layer of protection.
Robertson said so far 700,000 New Zealanders had signed up for My Covid Record, meaning they will be getting their vaccination certificates soon.
Vaccination certificates would be landing soon, he said.
The Government had paid out $5.5 billion in financial support during the Delta outbreak.
He said he had heard tales of retail bouncing back since it reopened this week, and in other places such as the Auckland CBD of retail doing it tough. He said that was why the Government's financial support was based on how much a business' revenue was down.
Robertson said a member of the Stratford family travelled to Auckland to pick up a family member and bring them back to Taranaki.
He said one of the six people had been vaccinated.
Robertson said one of the family declined a request to be tested earlier in the week but as a member of the family became sicker that resistance fell away.
None of the family have been using the Covid Tracer App.
Robertson said that at this time the advice was not to put Taranaki into alert level 3.
The reason for that is because the Government knows how the virus came to arrive in Taranaki and the family is in isolation. However, as more interviews are undertaken, the Government could change its approach.
Robertson said the health system is set up to make sure it does early assessments of whether people are safe to isolate at Covid at home. He said a standardised approach might not work for everyone in the community. That's why an individual assessment is needed.
Robertson denied the system was overwhelmed.
"We've got some examples where things have not gone the way we want them to go."
McElnay said there are constant meetings across DHBs to ensure they continue to learn how to deliver better care.
On the Auditor-General's report into saliva testing that showed some members of the panel choosing which provider to use had conflicts of interest, Robertson noted the Ministry of Health was acting quickly at the time and is confident it will now use the recommendations to guide their future processes.
Robertson said he wasn't involved in how the panel was chosen.
He said conflicts of interest did not automatically mean a person could not be involved in a selection process. Instead, they needed to be managed.
The current approach of making a phone call to Healthline as the first step in seeking help might not work for everyone, McElnay said and the ministry was keeping an eye on that.
Robertson said the Government's approach to putting a freeze on rent rise during last year's first Covid lockdown was not used this time because it was confident it had other measures in place to support those in financial trouble.
Robertson said he believed the person who left Auckland to travel with another family member to Stratford in Taranaki was the one person among the six new Covid cases who had been vaccinated.
Robertson said the Government still had the target of every DHB reaching 90 per cent of its eligible population but it also had to balance its other promise to allow Aucklanders to move around the country.
Where to get a vaccination in Auckland - without a booking
This map shows large vaccinations centres from the Unite again Covid-19 infomation page. For more detailed information about your neighbourhood visit Healthpoint.
He said it had made the commitment to let Aucklanders move around the country and so needed a system to back that up.
Whether that would mean an early move into the traffic light system for regions that hadn't been 90 per cent double-dosed, Robertson said he couldn't comment on it just yet.
However, the Government's decision on how this would work would be made soon, he said.
On the nurse's association speaking of severe staffing shortages, McElnay said she based her comment on the health system being able to cope with rising Covid numbers on her conversations with the heads of the DHBs.
McElnay said three of the new cases are aged under 12, but no schools had been identified as being locations of interest.
Robertson said the Government would make an announcement about the opening of the Auckland boundary next week.
The Beehive briefing this afternoon follows feisty nationwide debates about mandatory vaccines, the future of home isolation measures, and the country's "traffic light" health response.
It will not just be the numbers of any new Taranaki cases that may be consequential - but how closely any cases have followed health advice.
Six positive Covid cases in Stratford were very reluctant to get tested and have not been using the tracer app, a Taranaki DHB official told Radio New Zealand earlier.
The people who tested positive are a group of three adults and three children, who reportedly kept largely to themselves.
Meanwhile, Health Minister Andrew Little has admitted the system was struggling to cope with the high numbers of people with Covid-19 isolating at home.
At least three have died while isolating at home, most recently a man in Glen Eden. Their deaths have sparked debate over whether the MIQ system is functioning adequately.
Some health experts and Auckland Airport's chief executive have recently pleaded for major changes at MIQ.
The calls to allow low-risk, fully vaccinated returnees to skip MIQ have intensified in recent days, as the lack of space in quarantine compels many people to isolate at home.
Associate Professor Collin Tukuitonga said home isolation was unsatisfactory but sadly, surging case numbers meant more deaths were likely.
"Isolating people at home is a higher-risk strategy but enforced by the sheer numbers," Tukuitonga told the Herald earlier this week.
But vaccine mandates and the future of New Zealand's MIQ system remain contentious. And the Government also faces ongoing scrutiny of Covid-19 testing methods.
But what happens when you're packed and ready to leave but still don't have your test result? Former Cabinet Minister Maurice Williamson posed that question today, saying his wife was still waiting after 74 hours.
"There is going to be the odd example of a human error," Robertson said this afternoon.
But he said the average wait time for test results after taking swaps was currently 21 hours, which was satisfactory.
The system is a successor to the old four-step alert level regime, but is expected to rely heavily on district health board areas reaching the 90 per cent double vaccination target.