Health authorities on Wednesday said there had been a spike in the number of people attending special clinics set up around Adelaide, including a drive-through facility.

But they said unless people meet the criteria for coronavirus testing, which included a link to known cases or recent overseas travel, they should seek alternative medical treatment.

"I'd like to make it clear to people, we are going to look after your health needs no matter where we deliver that service," Health Minister Stephen Wade said.

"For people who meet the criteria, people who have travelled and who have the symptoms or who have close contact with a known carrier, we want to deliver that test in a more controlled environment, a dedicated clinic.

"If you present with symptoms but don't have those other criteria, we still want you to get the treatment you need but we ask you to go through your GP."

The minister's comments came as SA confirmed five more cases of the virus, including four linked directly to overseas travel, taking the state's total to 37.

Among those infected is a student at Flinders University, Vice-Chancellor Colin Stirling confirmed.

"A third-year student tested positive for COVID-19, but the student has not been on campus for eight days and has not been in direct contact with any staff or other students during that time," he said in a statement.

The professor said the student's partner, who had recently returned from overseas, was confirmed to have COVID-19, at which point the student self-isolated.

Close contacts of the student will be identified and contacted but it is understood no Flinders University staff or students are affected.

Despite the increase in cases, Deputy Public Health Officer Michael Cusack said he was still confident there was no widespread community transmission in SA.

"At the moment we are not seeing the background rates of virus in the community that clearly would suggest there is community transmission," Dr Cusack said.

"We are working very hard to delay that occurring in South Australia. Though a point we have made is we are planning for a time when we do see community transmission."

Also on Wednesday, Premier Steven Marshall dismissed suggestions of a complete state lock-down as "ludicrous" and he assured South Australians the state had adequate food supplies.

"There is no state lock-down, there is no CBD lock-down, this is wrong and it's unhelpful and it's completely and utterly ludicrous," Mr Marshall said.

But he said tougher restrictions on travel and gatherings in large groups were likely to remain in place for six months or more.

Mr Marshall also backed calls from Prime Minister Scott Morrison for people to stop panic-buying food.

"We're not going to run out of food in Australia. In fact, if there is any country in the world that can provide its own food security, it's Australia," he said.

Australian Associated Press