A worker at Newmont's Granites Mine spent seven hours at Alice Springs airport on Friday before flying to South Australia.
He then infected four family members with the virus in Adelaide, Northern Territory Chief Minister Michael Gunner said.
"Like all our other decisions we will not take a punt on this," Mr Gunner told reporters.
"We will operate on the assumption he was infectious while in the territory."
More than 25,000 people in Alice Springs were ordered to lock down for 72 hours from 1pm local time on Wednesday.
"We are still in a dangerous period. The territory is still under threat," Mr Gunner said.
The health direction applies to everyone inside the Alice Springs town council boundary, including hundreds of Indigenous Australians living in camps.
"Rough sleepers are among the most vulnerable," Mr Gunner said.
Chief Health Officer Hugh Heggie delivered a message in the Aboriginal Arrernte language of the Indigenous Alice Springs people.
He said to stay put, and that food, face masks and blankets would be provided.
"We're working closely with the (Central Australian Aboriginal Congress) and the council to make sure you're safe," Dr Heggie said.
The crisis started on Saturday when a young Victorian man, who travelled to the mine on June 18 via a Brisbane quarantine hotel, tested positive for the virus.
Hundreds of fly-in, fly-out mine workers travelled to NSW, Western Australia, Queensland, South Australia, Victoria, Tasmania and New Zealand in the days after he arrived.
There are so far understood to be 16 cases linked to the outbreak.
Two infected workers were diagnosed at the mine and evacuated to The Centre for National Resilience at the Howard Springs quarantine facility near Darwin.
Another mineworker, who had travelled to NSW, was found to be positive for the virus, as were two co-workers who travelled to Queensland, along with a close contact of one of them.
A Darwin man in his 50s, who left the mine on Friday, also returned a positive test.
He travelled to multiple venues, including Darwin's Buff Club, for more than four hours before being ordered to quarantine.
Another mine worker and his wife and daughter also tested positive after the mine worker had travelled to their family home in Palmerston, 20km south of Darwin.
They were quarantining at the Howard Springs facility when diagnosed.
Darwin and its surrounding areas have been in lockdown since Sunday, with hopes restrictions will end on Friday after wastewater testing found no evidence of COVID in the community.
