It was suggested below by a commenter that I should investigate movies with a Plymouth Fury featured in the film. After studying the design details of the Plymouth Fury I found that it does not have the S-shaped front vent window.
After looking for other late 1950s-early 1960s automobiles I found that all GM sedans of the years 1959-1961 had the curved front vent window. This includes Chrevrolets, Buicks, Pontiacs, and Cadillacs. Therefore I'm adding this information to the question which should now be widened to movies with a 1959-1961 sedan in them.
While I haven't found the answer to my question, here is s sample of something that has many of the elements of the piece I'm seeking. The title is "Night Fright" from 1967. The automobiles are newer than the ones I reference below, but there are many scenes of them being driven in daylight and night. There is a large, ambiguous monster lurking in the area as is typical with this genre. The look and feel of this color film is very similar to the piece in question. Viewing this particular film may dislodge some memories of other films of the same era that could be a match.
Having allowed for the elapse of some time and allowance for the recollection of faded memories I began to recall a skeleton of a story for this movie:
There as a hostile alien presence on earth (probably in S. California as usual) and the driver of the car that crashed had been attacked by a small device or creature. The driver was en route to tell others about this when the same attacker (that may have been hidden in the car) resumed its attack causing the driver to lose control. The sudden cutaway to the people in a the house was a scene where they the driver's wife and her friend were discussing strange recent events.
That is all I can glean at this point but I think it shows that the movie was of the alien invasion type and used the typical device of aliens preventing the facts about them from surfacing. For me, the details of the car, its crashing and the calm scene immediately afterwards are the defining moments of the movie, the rest of which probably featured a zucchini-shaped alien monster (or of some other absurd shape) that was eventually killed.
I still retain memories of a movie or TV show I saw in early childhood which may be of the science fiction or horror genres but I can only recall two distinct scenes. I am hoping that some details from these scenes will spark recognition in one or more members here.
The 1959 Chevrolet Impala
The first scene I recall featured a 1959 to 1961 vehicle, most likely a 1959 Chevrolet Impala.
The car could also have been a Buick of the same year or possibly another make from 1959 to 1961. The production year of the car is very important to this ID since cars with a curved front vent windows were only made in 1959 through 1961. The car in question had the celebrated curved front vent window.
Why is the curved front vent window so important?
The car in this movie or TV episode was being driven at a high speed along a curving road with a steep embankment at night. The driver lost control of the car along a curve and went off the road and crashed. The camera view immediately went to the driver side front window of the crashed car and showed a considerable amount of fake blood splash onto the curved vent window. The blood looked like ketchup when I first saw it and I still remember it being thick like the condiment.
The above was the first scene as I recall it in its entirety.
Two women talking
The second scene was an immediate cut away from the scene above to the interior of a house of the same period with two women, possible middle-aged and with dark hair speaking to each other about something related to the person involved in the car crash.
That's all I have for this inquiry unfortunately. No doubt there are a limited number of movies from the period of 1959 (the earliest possible year for the film or episode) through the mid-1960s showing two such scenes juxtaposed. The contrast between the two scenes was both vivid and jarring, especially for a young child and is likely why I still recall them clearly over 40 years later.
What I saw and recollect were scenes indicating that the movie or episode in question were part of a science fiction or horror movie of that period. I have seen so many movies of that type and from that era since then that I can now easily intuit their classification through only a brief viewing of them without any foreknowledge, so I know my inquiry is being presented in the correct forum.