The so-called Parafield cluster remains at 23 confirmed infections with three people in hospital in a stable condition.
However, Chief Public Health Officer Nicola Spurrier said there were 17 more suspected cases.
She said as a result of contact tracing and testing, 3200 close contacts of known cases were in quarantine and thousands more were self-isolating.
"With all of the 23 cases and the 17 suspected cases, what we are doing is putting a double ring fence around all of those people," Professor Spurrier said.
"So if you imagine all of those people have had close contacts and then those people have had close contact, that's where we're up to.
"Finding all of those people and requiring they quarantine."
Premier Steven Marshall said the statewide shutdown remained absolutely necessary to break the chains of transmission and prevent more widespread community infection.
"The lessons of surging infections in Victoria and other parts of the world have been learnt," Mr Marshall said.
"Indecision plays into the hands of this virus.
"COVID-19 is highly infectious, extremely dangerous and very difficult to eradicate once it gets a foothold in a community.
"So we need this circuit-breaker, this breathing space for a contact tracing blitz."
Mr Marshall said SA's contact tracing teams were working around the clock to track down people who might have been infected.
He also praised the thousands of South Australians who flocked to testing stations over the past two days, with more than 20,000 swabs taken as some people waited up to 12 hours.
Health Minister Stephen Wade said two more testing stations would be established to cater for the extremely high demand, one in Adelaide's north and one in the western suburbs.
Adelaide's city streets were largely deserted on Thursday as the lockdown took effect, with Police Commissioner Grant Stevens calling on people to do the right thing.
"This is a six-day period. We want people to honour the intention of this direction," he said.
"Please don't manipulate it to suit your own purposes."
Under the terms of the lockdown, all schools are closed along with universities, pubs, cafes, retail stores, food courts and takeaway food outlets.
Regional travel is banned and aged care centres are in lockdown.
Factories are closed, along with the construction industry, and elective surgery has ceased.
Weddings and funerals are banned as well as all outdoor sport and exercise, and masks are required outside the home.
People who are not essential workers will only be allowed to leave their homes once each day to buy groceries or to seek a COVID-19 test or other medical treatment.
Supermarkets, petrol stations, medical centres, critical infrastructure, public transport, airport and freight services, banks, post offices, school and child care for essential workers and veterinary services are open.