"The Hitch-Hiker" is the sixteenth episode of "The Twilight Zone" and was first broadcast in January of 1960. It was written by Rod Serling, and based on a radio play by Lucille Fletcher. It starred Inger Stevens as vacationing woman Nan Adams and Leonard Strong as the titular hitch-hiker. It also has a small role by Eleanor Audley, a famous voice actress.
We meet Nan Adams as she has started out her cross-country vacation. She has had a minor traffic accident, and the tow truck driver comments that she must have been lucky because the accident looks like it should have been worse. (Veteran or even occasional Twilight Zone viewers will probably say "uh-oh" at this point.)
As she leaves the scene of the accident, she sees a man waiting for her. She thinks nothing of it, until she seems him further down the road. As she crosses the country, the man is always shadowing her, but when she mentions it to others, they seem to not be able to see the mysterious hitch-hiker. (Aforementioned viewers are doubling their "uh-ohs" at this point). Trying to return to a sense of normalcy, Nan calls home only to have a shocking revelation that provides the twist ending to the story.
The milestone here is that this is the first Twilight Zone episode where the protagonist is female. And it might be seen as somewhat problematic that the first Twilight Zone episode to feature a female protagonist is a story about her being stalked by mysterious forces. But then, being stalked by mysterious forces is hardly an unusual circumstance in The Twilight Zone. The gender difference doesn't seem to have that much effect on the plot, other than a few places where customs seem a bit quaint. Otherwise, this is a well-told story about a mysterious threat that is not as threatening as it at first appears.