"What we know about clusters is that they grow," Bloomfield said.
"What is important is that we investigate these cases to their full extent, and that is exactly where people who are coming forward to be tested are helping us."
The new cases
Three of the new cases were employees at coolstore Americold, where one of Tuesday's cases also worked. Seven were family members of these employees.
One of the new cases is a Finance Now worker. Their family member is also infected.
The thirteenth case is a Mt Albert Grammar student who is a relative of the original four.
"The school has sent a letter to over 100 students and a handful of staff who are considered close contacts, advising them to self-isolate for 14 days," Bloomfield said.
"The student was not symptomatic while at school and has not been at school since they became unwell and got tested, so the chance of exposure to the school community is considered to be lower risk."
The new cases include other children. One is a girl aged 1 to 4 while the other is a boy between the ages of 5 and 9. Another girl who tested positive is aged between 10 and 14, while a teenage male (age from 15-19) is among the new cases.
There is also one probable case of Covid-19 in the community, also linked to the new outbreak.
Bloomfield revealed one of the new cases had visited an aged-care facility in the Waikato over the weekend.
Bloomfield said he wouldn't name facility yet as family members were still being notified.
But Health Minister Chris Hipkins said it was in Morrinsville and more details would be released shortly.
"There are a lot of rumours flying around at the moment. Everyone (at the rest home) who need to have been tested have been tested."
He refused to say if any were showing signs of symptoms, saying this would be released tomorrow.
The visit was when the person wasn't displaying symptoms but they developed symptoms the next day.
The visit was to a family member resident and the visitor saw their family member and was in contact with several staff.
Bloomfield thanked the facility for their response.
It comes after Aged Care Association chief executive Simon Wallace said yesterday all New Zealand rest homes had moved to alert level 4, meaning strict lockdowns and barring family visitors.
"Rest homes are on level 4 lockdown for at least three days. Full lockdown means effectively we are operating at level 4. It means there will be no family visits to their loved ones in a rest home," Wallace said.
"We've seen what's happening in Melbourne where aged care has been affected. More than 100 rest homes in Melbourne have had Covid-19 outbreaks and they didn't move quickly enough. We can't afford to take any risks at all."
Bloomfield said although all aged residential-care facilities were closed to the public, anyone who has visited the facility recently should also closely monitor their health and seek advice if they had any concerns.
Waikato DHB have confirmed they are today testing all staff and residents.
The facility has isolated the area visited from the remainder of the facility.
The DHB has staff on standby in the event that they're needed if the staff need to be stood down.
The DHB has ensured that the facility has all the PPE it needs.
Six rest homes had Covid-19 cases during the previous outbreak, two of which - Rosewood Rest Home in Christchurch and CHT St Margaret's in Auckland's Te Atatu - accounted for 16 of the country's 22 Covid-linked deaths.
An independent review found delays in realising Covid-19 was spreading within some rest homes meant more people were infected.
Tuesday's cases
Bloomfield said some of the original family who had tested positive were in a quarantine facility but not all of them.
Tuesday's outbreak was discovered after a man in his 50s went to his doctor with symptoms and was tested. Three family members then tested positive.
One of the four people worked at Mt Wellington cool store facility Americold while another worked at Finance Now.
Family members had travelled to Rotorua and Taupo over the weekend.
Bloomfield revealed further details about the trip to and urged anyone who was in these places at the same time to get tested.
On Saturday the family checked in at 3.30pm to the Wai Ora Hotel in Rotorua.
They visited the Herbs and Spice Thai Restaurant at 8pm that evening.
The following day, the Lakes District Health Board said the family visited the Fat Dog cafe on between 1.30pm and 2.30pm.
At 2.30pm they visited the Pak n Save supermarket at Amohau St, before heading to Heritage Farm and 3D Art Gallery at 3.30pm.
At 4pm they then headed to the Skyline Gondola and Luge.
The following morning, on the 10th, they visited Sail Barbary Eco Sailing in Taupō at 10.30am before returning to the Heritage Farm and 3D Art Gallery at 3pm.
There was also a visit that evening at 7pm to Burger Fuel Redwood Centre in Rotorua. On their way home, the family stopped for fuel in Hamilton at 9am on Tuesday morning.
The full list
The 16 cases below were listed on the Ministry of Health's website. But there are 17 cases of community transmission.
• Man aged between 50-59, Auckland DHB
• Girl aged between 1-4, Counties Manukau DHB
• Woman aged between 50-59, Counties Manukau DHB
• Woman aged between 60-69, Counties Manukau DHB
• Boy aged between 5-9, Counties Manukau
• Man aged between 30-39, Counties Manukau
• Man aged between 40-49, Counties Manukau
• Man aged between 50-59, Counties Manukau
• Man aged between 50-59, Counties Manukau
• Girl aged between 10-14, Waitematā DHB
• Boy aged between 15-19, Waitematā DHB
• Man aged between 40-49, Waitematā DHB
• Woman aged between 20-29,Counties Manukau DHB
• Female aged between 50 to 59, Counties Manukau DHB
• Boy aged <1, Counties Manukau DHB
• Man aged between 50-59, Counties Manukau DHB
