It would be missing its right wing mirror, as this broke off when the cyclist was struck.
The vehicle was heading away from Hamilton at the time and continued on in the westerly direction after the crash.
Senior Sergeant Rupert Friend of the Hamilton District Command Centre said: "Police are very keen to hear from anyone who can assist us in identifying this vehicle and its driver. We are extremely lucky that we are not investigating a fatal crash on this occasion."
This is the second hit-and-run that police have been chasing today.
The second collision happened in Rotorua about 10:25am.
The car, described as a grey coloured hatchback, was last seen driving away from the accident with a motorcyclist towards Waterford St, police said.
The motorcycle rider was left with moderate injuries and was transported to Rotorua Hospital.
Anyone with information is asked to call Crimestoppers anonymously on 0800 555 111.