Both of the latest two people to die of Covid-19 were elderly, had underlying health conditions and could be linked to existing clusters.
One was a man in his 80s who passed away at Wellington Public Hospital yesterday. He first became unwell on March 26 and was admitted to hospital two days later.
The man's infection was linked to an existing cluster which McElnay said they wouldn't identify for privacy reasons.
"We are confident that his close contacts have been traced."
The second was a man in his 70s who died in Burwood Hospital, Christchurch, after being among the group transferred from the Rosewood Rest Home.
McElnay said because of the "underlying vulnerabilities" of that group, further deaths or serious illness couldn't be ruled out.
"We acknowledge the anxiety New Zealanders may be feeling about today's news, both in the wider community and also for family and whanau grieving over these sad deaths."
Because of the level 4 alert, McElnay said they understood how difficult it was for families and whanau to feel they are part of their loved ones' final hours.
On behalf of the Ministry of Health, McElnay thanked the frontline health workers who'd been providing patients with comfort and support during these times.
"Again, today's news reinforces the importance of our move to Alert Level Four, and the measures we are all taking to limit spread, break the chain of transmission and prevent deaths.
"We want you to know that our health system will continue to do everything it can to support the fight against Covid-19."
New Covid-19 cluster in Christchurch
Three of the four deaths in New Zealand are linked to clusters.
There are now 13 clusters of Covid-19 across the country.
The George Manning residential care facility in Christchurch is this country's 13thcluster where more than 10 people have been infected with Covid-19 from a single source.
One of the clusters is linked to a private party in Auckland while another cluster can now be linked to Spectrum Care - a daycare facility for children with disabilities.
McElnay couldn't give the date of the private function in Auckland.
She said there was ongoing transmission happening from people at the function.
McElnay said they hadn't missed an opportunity to contact tracing by not naming the Auckland clusters because the contact tracing was ongoing behind the scenes.
McElnay said family members of the Rosewood rest home should contact the local DHB directly for answers about what to do.
The investigations into how people at the Christchurch rest home were infected are still ongoing.
McElnay said the Canterbury DHB did the right decision to move the patients to provide the "extra level of support".
At the George Manning rest home where there are 14 cases, there was a mixture of residents and staff and McElnay said the virus was "well contained" there.
At Rosewood, McElnay said the group was "frail and elderly" and the impact of Covid-19 was that impacted that group could be a "very unpleasant illness".
McElnay said the would tell all the DHBs today to contact all of their aged care facilities and tell them about what procedures they should already have in place.
Frontline workers test positive for Covid-19
McElnay said there was now three nurses who had tested positive from Waikato Hospital and investigations were ongoing but the nurses were all close contacts out of the workplace.
New Zealand's second confirmed death from Covid-19 was confirmed yesterday after a 90-year-old woman with underlying health conditions passed away in Burwood Hospital.
She was one 20 residents hospitalised from the Rosewood Rest Home after someone tested positive at the facility.
The other death was West Coast woman Anne Guenole, 73, who passed away in hospital on March 29, two days after testing positive for Covid-19.
Meanwhile, a group of 16 New Zealanders and 96 Australians will board a repatriation flight in Uruguay this afternoon.