The incident took place shortly after 1.15am on Sunday October 3, on the A720 westbound near the Lothianburn Junction.
A 31-year-old man had stopped his Volkswagen car on the on-slip road to repair a puncture when he was struck by a vehicle.
Enquiries have indicated that the vehicle which struck into the Volkswagen was a white Audi A4 or S4.
The car failed to stop and continued on the City Bypass in the direction of the Dreghorn Junction.
The 31-year-old was taken to the Royal Infirmary of Edinburgh for treatment to serious injuries.
Police Scotland has said that their investigation into the incident is ongoing and are asking anyone who has seen the vehicle which collided into the Volkswagen to get in touch.
Inspector Roger Park, from Edinburgh City Road Policing Unit, said: “Our investigation into this crash indicate that the vehicle involved is now believed to be a white Audi A4 or S4.
"It would have obvious collision damage to the front bodywork.
“We are asking for anyone who believes they may have seen a vehicle matching this description or something similar to get in touch.
“We would also be keen to speak to anyone who was on the City Bypass early on Sunday morning and may have dashcam footage to come forward.
“Anyone with information can call 101, quoting incident 0282 of 3 October.”