The missing shoe of a man dumped by a couple at a Salisbury North home in Adelaide north suffering serious head injuries has been found about 25 kilometres away at Torrensville.

SA Police still have not been able to talk to the 30-year-old man, who is in the Royal Adelaide Hospital in a critical condition.

Police are appealing for information about the man's movements and the couple who dropped him off at the Deering Street home of "an associate" on Wednesday night.

Detective Chief Inspector Shane Addison said the last confirmed sighting of the man was about 5:00pm at the corner of Edgecombe Road and Peachey Road at Davoren Park.

The man was wearing a fawn-coloured t-shirt with the number 65 on it, blue jeans and dark blue DC brand sandshoes, he said.

"One of the sandshoes was initially missing, but was found last night on Henley Beach Road at Torrensville after a member of the public called police after seeing the pictures," he said.

"Police investigations have now confirmed the victim was with a blue or green coloured vehicle stopped on Henley Beach Road, Torrensville between 8:00pm and 9:00pm."

Chief Inspector Addison said the man's injuries were not consistent with being hit by a car.

Anyone with information has been urged to contact Crime Stoppers.