South Australia has reported five local COVID-19 infections, all involving the family of .
But Premier Steven Marshall says the state will not be going into lockdown.
The new cases involve the miner, his wife and three of their children, all aged under 10.
"While this is a very concerning turn of events, we are very relieved this person and this family have been at home since Saturday," Mr Marshall said.
He said the family had all been transferred to Adelaide's dedicated facility for positive coronavirus cases.
The cases were regarded as "linked transmissions" from the Northern Territory exposure mine site.
The other 28 miners who returned from the same NT mine have also been in isolation since arriving in SA and are being re-tested.
Chief Public Health Officer Nicola Spurrier said officials had moved to require anyone on the same Alice Springs to Adelaide flight as the infected miner on Friday, to go into isolation.
More than 100 people who were on board VA1742 have already been contacted with a small number still to be notified.
Checking is also underway to identify any possible contacts at Adelaide Airport and at a suburban takeaway food outlet the man visited on his way home.
Both those locations are considered low risk.
Professor Spurrier said while the outbreak was regarded as contained at this stage, more work remained and officials did not want the Delta strain of the virus to become widespread in the community.
"What we're seeing with this Delta variant is there's almost a 100 per cent hit rate for close contacts," she said.
In response to the new cases, SA's transition committee has reduced the number of people allowed at family gatherings to just 10.
While the AFL game between the Adelaide Crows and the Brisbane Lions at Adelaide Oval on Saturday is still expected to go ahead, a decision is yet to be made on whether or how many spectators will be allowed.
Mr Marshall said people were being advised to work from home where possible, and everyone was urged to use QR codes, get vaccinated and observe social distancing.
"We can see the consequences of getting this wrong. We don't want a lockdown in South Australia," the premier said.
Amid reports of panic buying across SA supermarkets, the premier said there was no need for such action.
