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The Ministry of Health is advising anyone who stayed overnight on Saturday in the Green Lake Hut to self-isolate and get tested immediately.
Officials regard the remote wilderness setting as a high-risk for anyone in the same venue, the latest locations of interest show.
A close contact of an infected person had arrived at the hut on Saturday at 7pm and left the following morning at 8am, the ministry said.
Green Lake Hut is a three to six-hour walk on the Green Lake Track, which is considered an advanced or challenging hike.
According to the DoC website it has 12-bunk beds and bookings are not required, operating on a first come, first served basis.
Earlier this month it was announced that from December 15 all hikers over the age of 12 will have to be fully vaccinated against Covid to stay in DoC accommodation and campsites.
"All DoC campgrounds and huts will only be open to those who are fully vaccinated. When booking, visitors will be required to confirm they and all others in their group are fully vaccinated," DoC Heritage and Visitors Director Steve Taylor had said.
"Hut wardens and camp hosts are regularly on site and will be checking for vaccination status."
In more remote regions and huts without wardens there would be spot checks for campers' My Covid Pass.
Meanwhile a second staff member at Enner Glynn School has tested positive for Covid-19.
In a post to Facebook, principal Isaac Day said they received confirmation from the Medical Officer of Health about the new case.
The staff member is considered to have been infectious from December 4, it said.
"As the staff member was in home quarantine during his infectious period, they are not thought to pose any risk of infection to students and the community.
"We ask please that all in the school community maintain extra vigilant at this time. In particular, if anyone develops symptoms that might be Covid...please don't come to school, get tested, and self-isolate until you have a negative result."
He said the school continues to take all necessary precautions to keep children safe.
It comes after it was revealed last week, two staff members, one at Enner Glynn School and the other at Broadgreen Intermediate, had tested positive for the virus.
The schools were closed for a number of days last week.
And there were four new cases of Covid-19 to report in the Nelson-Tasman area yesterday. It brings the total number of active cases to 15.