Health and Education Minister Chris Hipkins says the Rydges worker who tested positive received only a temperature check while still working with a cough because it was the "regular system of testing".
The man had attributed a cough he had for two days to an existing condition.
Temperature checks on managed isolation and quarantine facility staff are taking place every day, Hipkins has told Newstalk ZB's Heather du Plessis-Allan's Drive show.
If people then are showing symptoms, they will be Covid-19 tested.
The man is not linked to the existing Auckland August cluster which has 98 people linked to it; all of whom are in quarantine.
The worker tested positive on Sunday, August 16, but showed symptoms from August 11.
People working at the Jet Park Hotel will be tested each week, Hipkins said.
Not every worker at the border is being tested and it would be unnecessary to do so, he said.
People working on borders who are showing symptoms will be tested for Covid, he said.
Asked whether all air crew are being tested coming into the country, Hipkins said they were testing those from "higher risk" countries.
Hipkins will issue an order later this week outlying how air crew will be tested.
He says there is "a lot of complexity" around the testing and that he was working with airlines to sort out any logistical issues.
"We've got to understand the risk, all of the potential areas where things can go wrong and mitigating them," he said.
All front-facing border staff - a couple of hundred workers - will be tested each week, Hipkins said.
Maintenance workers, like the man who tested positive without a link to the Covid-19 cluster, would not be classed as a front-facing worker.
The man was transferred to the Jet Park Hotel quarantine facility yesterday, the Ministry of Health said in a statement before the usual 1pm press conference.
Contact tracing and testing have not connected any further cases to the worker.
However, he does have close contacts. He attended two Emmanuel Cook Islands Good News Fellowship services on the morning and evening of August 9.
As of last night, all but nine of them had been tested and the remainder will be tested today. All are in self-isolation.
Deep cleaning of the school hall where the service was held is under way.
Six close contacts he worked with at Rydges have been identified and are all self-isolating, the ministry says.
All returned negative results from surveillance testing last week, however, as a precaution all staff and returnees at Rydges are being retested again.
To date, the maintenance worker remains a single case.
Today it was announced there are 13 new Covid cases in the community.
12 positive cases are linked to the current cluster and one remains under investigation but is believed to be linked.
98 people linked to the cluster, including 44 positive cases, have been moved into quarantine facilities.
There are six people receiving hospital-level care. Two are in Auckland City Hospital on a ward, and four people are in Middlemore.
The total number of active cases in New Zealand is 90, of which 69 are from the recent cluster, one more is a case in the community, and 20 are imported cases in managed isolation and quarantine facilities.