The 31-year-old was walking the pavement on Bridge Street, Walsall, when he was hit by a silver VW Golf .
The man suffered a "nasty" head injury but is now recovering after the incident which took place at around 4am on Saturday.
The vehicle didn't stop after the collision and collided with bollards in George Street shortly afterwards before heading towards the Caldmore area, West Midlands Police said.
It was seen driving dangerously around the town centre, including on Freer Street and Goodall Street, with the force now urging people to come forward.
Nobody has been arrested over the incident and the car is yet to be found.
A spokesman for West Midlands Police said: "We are asking anyone with dashcam or phone footage of the incident to please get in touch.
"We are particularly looking for sightings of any silver VW golf cars in the area, similar to the one pictured, with damage to the front end.
"Please speak to us via Live Chat on our website west-midlands.police.uk and quote log 635 of 11/09/21."