In the beginning, Eustis Clay is seen admiring a parked sports car. It is a 1962 or early 1963 Shelby AC Cobra, one of the first cars Carroll Shelby made, and extremely valuable.
The large patch on the left shoulder of everyone's uniform is that of the U.S. Sixth Army. At the time of this film, it was headquartered at The Presidio in San Francisco, CA, and was used for training soldiers being sent overseas. Inactivated in 1995, it was reactivated in 2008 and is now headquartered at Fort Sam Houston in Texas.
Jackie Gleason released a version of the title music as a single on Capitol in early 1964, with another Henry Mancini composition, Bird Brain, as the B-side. This puts "Soldier in the Rain (1963)" in the same category as "Thunder Road (1958)" and "Because They're Young (1960):" movies which had a version of the theme song performed by a member of the cast and released as a single, while that version wasn't the one used in the movie.
The camera Clay uses for the "pin-up" contest is a Polaroid Land Camera Model J66. It uses Type 47 black-and-white roll film and was manufactured between 1961 and 1963. Retail price was $89.50 (or about $700 in 2016).