The car that they take in trade from the mechanic is a four-door 1959 Plymouth Belvedere. This car, with its distinctive tail-fins, is in the same family of cars as the Plymouth Fury, made famous in the movie Christine (1983).
Three of the characters here are listed in the credits simply for the title of their position in law enforcement, but as revealed in the story, the Texas Sheriff, played by Edward Faulkner, goes by the name Frank Brill; the Police Sergeant in Iowa, played by Hal Lynch, goes by the name Jay; and the New Mexico Deputy, played by Robert Sorrells, goes by the name Hank.
The journey Dr. Kimble takes in this show passes through Tucumcari, New Mexico, on the historic Route 66. Tucumcari has loomed large in popular culture for decades, serving as a venue for movie and TV series shoots, from Rawhide (1959) to The Ballad of Buster Scruggs (2018). It's also prominent in the lyrics of Lowell George's song "Willin'."
This is the second and final appearance Suzanne Pleshette makes in the series, each time playing a character who becomes romantically interested in Dr. Kimble.