Joan Price, 59, from Troon, died and another woman was seriously hurt in the crash on the A77 near Ayr on Monday.
The driver of a second vehicle involved in the collision left the scene of the accident near the Holmston roundabout before police arrived.
Officers believe the man and his passenger escaped in a silver Audi A3 which was on the A77 at the same time.
The black Volkswagen Golf in which they had been travelling was "undriveable" following the crash, Sgt Kevin Blackley said.
He added that, given the extent of the damage to the car, it was likely that at least one of the men was injured.
The Golf was travelling southbound on the A77 when it hit the northbound red Nissan Pulsar at around 21:35.
Joan Price, who was driving the Pulsar, died at the scene. Her 39-year-old female passenger was taken to Ayr Hospital with serious injuries. She is said to be in a stable condition.
Police collision investigators closed the road between the Whitletts roundabout and Holmston roundabout for several hours following the crash.
Sgt Blackley said: "From witnesses, the two men in the VW Golf got out of the car and jumped into a silver Audi A3 that had been in the vicinity at the time of crash.
"The VW Golf was undriveable and recovered at the scene. Due to the extent of the damage to the car, it is possible that either one or both of the occupants could have been injured by the impact of the crash.
"A number of people came to the aid of the two women in the car and called the emergency services. However, we still would like to hear from anyone who was on the A77 around the time of the crash and who saw it happen.
"We are particularly keen to speak to anyone who may have seen the cars - all three - prior to the crash and who can provide information about the way they were being driven at the time - this can be either on the A77 or on the nearby roads.
"Anyone with information that may assist our investigation should contact the divisional road policing unit at Irvine through the 101 number or anonymously through Crimestoppers."
Police are hunting two men who fled the scene of a fatal road accident in Ayrshire.