There are now 20 probable or positive cases linked to the cluster in northern Adelaide.
State health officials have embarked on a testing blitz with hundreds of people lining up at pop-up clinics.
Australia's deputy chief medical officer Michael Kidd said the next 24 to 48 hours would be critical.
"Health authorities in South Australia are doing a terrific job, identifying people who may have been in contact with someone who's been diagnosed with COVID-19, arranging to get people tested, arranging for people to go into isolation while they're waiting for their results," he told the ABC on Tuesday.
"And a big thank you to everybody in Adelaide who were queuing up to get tested yesterday, and I'm sure there will be queues today for people waiting to get tested for COVID-19."
The cluster could derail national plans to open state borders before Christmas.
All states except Western Australia have committed to easing restrictions by December 25.
But Queensland, the Northern Territory, Tasmania and WA have now slammed the gates shut on South Australia.
Victoria - which has now gone 18 days without new coronavirus infections or deaths - and NSW have increased screening for Adelaide arrivals.
Prime Minister Scott Morrison argues the changes should only be temporary and entire states should not be locked down because of an outbreak in one location.
Australian Medical Association president Omar Khorshid is less concerned about reopening borders by Christmas than he is about reopening society more broadly.
Dr Khorshid is concerned that once families can get together in big groups again, people will quickly forget the lessons learnt throughout the pandemic.
"We've just seen demonstrated very clearly in South Australia that we're only ever one quarantine breach away from it getting back into our society," he told the Seven Network.
"We need to make sure that we've got enough social distancing, enough restrictions in place, that any outbreak that does happen is small and able to be really quickly controlled.
We're going to need to stay in that situation for many, many months to come, even once vaccines are starting to become available.
More Australian Defence Force troops are on the way to help SA while international flights to Adelaide are paused for the rest of the week to ease the burden on hotel quarantine.
WA and Commonwealth contact tracers are helping state officials to stop the cluster from growing.
The state has re-introduced a range of restrictions, including gyms, recreation centres and play cafes closed for two weeks.