South Australia has recorded a further six cases of Covid-19, taking the cluster to 12.
"Unfortunately, we have new cases to report and concerning new exposure sites in addition to that," Premier Steven Marshall told reporters late on Wednesday afternoon.
The new cases include five linked to the Tenerife winery.
"We believe this was an exposure site on Sunday afternoon from 1.45pm through to 4.30pm," Marshall said.
"So five new cases linked there, and one additional case linked to The Greek on Halifax St."
Marshall said it was "most concerning" that one of the new exposure sites was the Gawler District B-12 school, and one of the teachers infected was at the site on Tuesday.
"We will be getting in contact with every person that we can who was at these exposure sites," he said.
"This is exactly and precisely why we needed to move South Australia into a lockdown situation."
Earlier, it was announced that eligible South Australians were being urged to book in for their Covid-19 vaccinations, after the Premier revealed hundreds of appointments available this week have not been filled.
Marshall said despite the slow take-up, there had still been a record amount of vaccinations administered in the state's clinics on Tuesday.
"We know vaccines are our pathway out of these stop-again-start-again lockdowns being experienced around Australia at the moment," Marshall said.
"Looking at the website, there are still about 700 appointments available over the next seven days at ... clinics.
"Please go onto the SA Health website to see if you are eligible for the vaccination.
"This is our pathway out."
Chief health officer Nicola Spurrier said the new case was linked to The Greek on Halifax restaurant.
Professor Spurrier urged anyone who had visited the restaurant between 6pm and 10pm who hasn't been contacted to call SA Health and let them know.
She said anyone who was present at the restaurant on Saturday night needed to enter isolation and get tested immediately.
The state's outbreak was sparked over the weekend by an 81-year-old man who completed hotel quarantine in Sydney after returning from Argentina with his daughter and flew to Adelaide on July 8.
He presented to Modbury Hospital on Sunday with symptoms before testing positive, putting the hospital's emergency department into lockdown.
Two of the man's close contacts – his daughter and a man, both in their 50s – tested positive for the virus on Monday.
The fourth case, a man in his 60s, was a close contact of the 81-year-old and was announced on Tuesday morning.
The fifth case is a man in his 60s who picked up the virus at The Greek on Halifax restaurant on Halifax St, Adelaide, on Saturday. His case was announced later on Tuesday.
Health officials have said they are most worried about the fifth case, who was infected at a birthday event at The Greek where another positive case was present.