South Australia's cluster of COVID-19 cases has jumped to 12, with fears two exposure sites could become superspreaders.

South Australia's COVID-19 cluster has jumped from six to 12 cases with another school added to concerning exposure sites.

Five of the new cases are linked to the Tenafeate Creek winery at Yattalunga, north of Adelaide, with the sixth, a child under five, linked to The Greek restaurant in the city where a number of infected people dined over the weekend.

One of the cases from the winery, a man in his 40s, also attended a school at Gawler on Monday raising concerns for other staff and students.

Chief Public Health Officer Nicola Spurrier said efforts were continuing to contact trace everyone at the winery and all those linked to the school.

"This just shows how quickly this can spread, how quickly people can get infected," she said.

"I would say both The Greek and this winery are examples of superspreader events.

"So this is very serious. I am concerned but I am also very pleased that we were able to put a lockdown in place very quickly."

As the seven-day lockdown continues and the number of exposure sites climbed well above 50, SA will bolster testing resources.

Premier Steven Marshall said the state would do everything it could to increase testing resources amid reports of long lines forcing some to wait for more than 10 hours.

Mr Marshall said officials were "pulling every single lever possible" to increase testing capacity with Defence personnel, both local and interstate, to help and interstate nurses and doctors to lend further assistance.

Changes would also include the opening of a dedicated facility for those people directed to get tested and enter home quarantine and the publishing of estimated wait times for all of SA's more than 80 testing sites.

"We don't want people angry and frustrated with delays. We are ramping up capacity across testing sites," the premier said.

"We've got one shot to stop this outbreak dead in its tracks.

"We've got a chance to nip this in the bud. We don't want to have an extended lockdown in South Australia."

SA's list of exposure sites includes major shopping centres and schools with almost 6000 people linked to those sites now in home quarantine.

SA's virus cluster is linked to an 81-year-old man who recently arrived in Australia from Argentina and quarantined in Sydney before travelling to Adelaide, where he tested positive.

Genomic testing has confirmed the man became infected while in Sydney, not while in Argentina.

During the week-long lockdown, which started on Tuesday evening, South Australians will only be allowed out for five reasons - to provide essential care, to seek medical assistance, to buy essential food and other goods, for essential work, or to exercise for up to two-and-a-half hours.

They must also wear masks when outside their homes.

Schools have closed along with most retail outlets and all construction work has ceased.

Police Commissioner Grant Stevens said the SA community was doing a good job in complying with lockdown rules.

"Every indicator we've seen so far is that people are doing the right thing," he said.

"Streets are empty, the city is empty.

"I'm as hopeful as anyone else that we can keep this lockdown to seven days, but there's a lot of work to be done."