The State Emergency Service says significant flooding is expected and may pose a risk to the safety of local residents.
South Australia Services Minister Peter Malinauskas warned the public to exercise more caution.
"It is unfortunate that some motorists haven't heeded the warnings of emergency services," he said. "There was indeed one vehicle that had four passengers in it that had to have a swift water rescue take place.
It's been "unrelenting", he added.
"This whole weather event has been incredibly dynamic, constantly receiving updates.
The areas most likely to be impacted include Hillier, Gawler River, Angle Vale, Penfield Gardens, Two Wells, Lewiston, Virginia, Port Gawler and Buckland Park.
The SES says people living in those areas should follow their flood emergency plans and if they plan to leave should consider leaving now.
The Gawler and North Para rivers are continuing to rise after 100mm of rain fell across the catchment areas over the past two days.
Flood warnings also remain in place for the Onkaparinga catchment area, south of Adelaide, and for the Torrens River which runs through Adelaide's northern suburbs and into the city.
However, improving weather across the state has allowed SA Power Networks to reconnect more properties after Wednesday's statewide blackout and ongoing issues with storm damage.
By Friday about 11,000 people were still without services because of the network failure, mostly on the Eyre Peninsula and in the far north.
Transmission company ElectraNet says ground crews are continuing to work on the line between Cleve and Port Lincoln to repair a fault west of Tumby Bay that only became evident during an earlier attempt to restore services.
It says some supply has been restored to Port Lincoln with the use of back-up generation and early indications suggest full supply may be possible at some stage on Friday.
It is also conducting aerial surveys of transmission lines to Roxby Downs and Leigh Creek and if no damage is spotted, attempts will be made to restore those services.
Concerns are growing for some regional centres where power has been out since Wednesday, cutting the supply of fresh food and access to fuel and cash.
The Bureau of Meterology has cancelled all severe weather warnings for SA but has forecast more light rain across the flood-affected areas on Friday ahead of a dry day on Saturday.
IVF patients lose embryos in power outage
Twelve families have lost embryos at an Adelaide fertility clinic hit by power outages.
Flinders Fertility lost power and its backup generators failed during the statewide blackout on Wednesday, compromising incubators.
"Despite every effort by our scientists, the embryos are no longer viable," the clinic said in a statement.
By Friday morning about 12,000 were still without services because of the network failure, mostly on the Eyre Peninsula and in the far north.
Transmission company ElectraNet says ground crews are continuing to work on the line between Cleve and Port Lincoln to repair a fault that only became evident during an earlier attempt to restore services.
It also hoped to survey lines to Roxby Downs and Leigh Creek from the air on Friday.
The intense low pressure system that smashed SA was packing winds of up to 140km/h, among the strongest the city has experienced, prompting an unprecedented warning from police for workers to head home early and stay home amid concerns emergency services might not be able to cope.
Mr Weatherill described the storm as "catastrophic" and said it had involved weather events not seen before in SA, "such as twin tornadoes, which ripped through the northern parts of our state".
If you need help:
For storm assistance, call 132 500
For life-threatening emergencies, call Triple Zero (000)
