South Australia's Department of Correctional Services said on Sunday that a communications fault with its monitoring system had led to its being unable to track 774 offenders.

However, all except one, had been accounted for, the Department said.

South Australia Premier Steven Marshall said an inquiry would be launched to find out why the outage had occurred.

A spokesperson from the Telecommunications Industry Ombudsman said small business customers affected by the outage may be entitled to compensation.

Businesses first needed to try and have the issues they faced resolved by Telstra, failing which they could complain to the TIO, the spokesperson added.

A Telstra spokesperson said a "complex issue" had caused the outage.

"The issue was caused by a fault in a vendor's network and we had expert technicians onsite to assist them with restoration," the spokesperson said.

"The issue was a complex one involving multiple suppliers across a number of locations … we worked through the [Friday] night to restore services as quickly as possible."