"Had this person been more upfront with us, we would not have instituted a six-day lockdown.
"This is the one element that pushed us from the level of restrictions we were implementing on Tuesday to a much harsher regime."
Asked if the man would be deported when his visa expired, Stevens said: "Whatever occurs in relation to his status as a person in Australia will be subject to the outcome of our investigation process."
Twenty detectives are working on the case.
Stevens said he and chief public health officer Nicola Spurrier were satisfied they made the right call at the time based on the information available.
"To be clear, the decision to lockdown hard ... was not based on the interview with one man," Spurrier said.
"We would never make those decisions in isolation with just one piece of information — it's very complex."
Spurrier said SA had recorded one new case of Covid-19 overnight, as SA prepared to end its six-day lockdown early on Saturday night.
"This is not an example of community transmission, or just something has just popped up out of the blue."
Spurrier said it was a close contact of a previous case who was already in quarantine, so they posed no community risk.
"He's been in quarantine since his partner tested positive.
"So it's definitely linked to the Parafield cluster."
There are now 26 cases connected to the outbreak and one person remains in hospital.
Spurrier said more than 5400 people linked to the Parafield cluster had been contacted and put into quarantine.
"We have made huge inroads into the investigation and getting on top of this outbreak."
There are 37 active cases in SA.
"This next week for us is critical in our state," Spurrier said.
"It's the time for us to search for that general community transmission."
Spurrier revealed there were about 40 close contacts who authorities had not been able to get hold of directly, saying that was "a big push" for Saturday.
"Everybody receives an SMS message to be told that you are a close contact and you must go into quarantine and get tested, but of course we want to follow that up and speak to people directly," she said.
Spurrier also revealed a junior doctor was responsible for picking up the state's first case, which sparked the Parafield cluster.
Marshall said 19,000 coronavirus tests had been conducted on Friday — up from the usual 2000 or 3000.
"This is an extraordinary response and for that I'm very grateful."
Marshall said it had been an extraordinary week.
"But my message to South Australia is very clear: we are not out of the woods yet.
"We are still managing a very dangerous cluster, and although we are reducing those restrictions, we are still very concerned about this cluster."
Marshall said while the risk had been "diminished" and "refined", there was still a very real risk.
