SA on Tuesday reported 3773 new cases - with 2087 of those detected by rapid antigen tests - and a total of 8336 PCR tests.
Given those results and the level of hospital emergency presentations, Health Minister Chris Picton said it was clear some people with COVID-19 were not getting tested.
"We know there are people presenting to emergency departments at the moment who don't know that they have COVID because they haven't got tested," Mr Picton said.
The minister said this not only distorted the state's virus statistics, but also had an impact on the early provision of anti-virals to people with an infection.
He said it was too early to tell if SA had reached the peak of the latest wave, sparked by an increase in cases of the new BA.4 and BA.5 Omicron variants.
But Mr Picton said the latest daily case numbers appeared to be in line with recent modelling.
That suggested a daily peak of up to 6000 cases, though SA has only gone beyond 5000 on one occasion so far.
On each of the past four days, they have dipped below 4000.
On Tuesday 379 people were in hospital with the virus, down from Monday's record of 384 but still well above the 216 in hospital a month ago.
Australian Nursing and Midwifery Federation SA chief executive Elizabeth Dabars said the situation remained extremely challenging for health workers with the system under significant duress.
She said the government needed to find extra beds and people to staff them. With about 1200 SA Health workers currently furloughed because of COVID-19, that was proving difficult.
Professor Dabars called on the wider community to continue to take their health and welfare into their own hands by voluntarily wearing masks and taking other precautions.
"Please do keep yourselves safe and healthy. In the event that you do need assistance and help, please make sure you support the people who are supporting you," she said.
"We do have a concern that with the increase in pressure on the system that the frustrations that the community experience will spill out over to our hardworking and dedicated health and nursing workforce.
"That's not what we want."
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