Since the two positive tests were returned, 300 people who worked at the centre had been tested.
While no additional positive tests had been returned yet, a further two people were unwell, NZ Post chief operating officer Mark Stewart said.
Health officials had now advised all those working on the day shift should now be considered "close contacts".
"This is an unsettling time for our people and their families," Stewart said.
"We are supporting those who are directly affected, respecting their personal situation and supporting other teams at NZ Post who are concerned about their colleagues and for each other."
The company had strict safety measures in place under alert level 3 and 2, he said.
"This includes 2 metre physical distancing, mask wearing and hygiene measures at the Auckland Operations Centre and we are very vigilant about following them. This will of course continue to be a priority for us."
After the two positive tests last week, NZ Post followed advice from health officials and conducted a deep clean of the site, stood down close contacts and additional controls including 2 metre physical distancing and other PPE controls.
The site was deemed fit to reopen after those actions were taken, Stewart said.
"We are again following health officials' advice and doing a further deep clean today. Officials have advised that the site is safe for our other processing shifts to keep operating, with our Covid safety measures in place."
They were still figuring out the implications of the shift change to processing and delivery times, but because of existing level 3 safety measures there could be up to two-day delays for items delivered to the Auckland area, Stewart said.
Delivery of mail was not affected at this stage, he said.
