The IMDB page on the movie Terminal Station has only 3 listed tracks:

Autumn in Rome Sung by Patti Page (link)
Indiscretion Sung by Patti Page (link)
Rhapsody in Blue

Another website I've never heard of (filmscoremonthly.com) gave the following track list:

INDISCRETION OF AN AMERICAN WIFE (1953) mono
1. Terminal Station 4:07
2. The Meeting 4:19
3. Rendez-vous 3:41
4. Search and Near Accident 3:21
5. Deserted Car 2:37
6. Decision 3:56
7. Parting 2:49

You can hear snippets of each track here and attempt to ID the song you heard, though most were pretty typical 50s flowery violin background music to me.

Minor tidbit, but it was copyrighted under the name Cal Bagby, a pseudonym of Lou Gottlieb.

So unfortunately I can't find much in the way of answers