Factual error: There seem to be modern "Walk" and "Don't Walk" signs at each intersection. This film is supposed to take place in 1952. Also it is mentioned there are "Stereophonic" consoles in the motel rooms. Probably would not find those until at best the late 50s.
Factual error: As they drive through the tunnel out of NYC the radio is playing. Radio waves can't travel in either the Holland or Lincoln tunnels.
Factual error: Carol talks about getting on the "interstate", but the interstate system wasn't even initiated until 1956.
Answer: The detective has likely been on Carol's tail following her for some time, not just recently.