http://en.wikipedia.org/wiki/ZIP_code claims without citation that zones were introduced in 1943. http://web.archive.org/web/20041030125608/https://www.usps.c... confirms May 1943. It's not clear when the envelope was supposed to be from, so it may be right that it doesn't have a zone like "37" after Chicago.

Awesomely, using a ZIP code on a supposedly pre-ZIP letter was one of the clues that exposed a document forger: http://en.wikipedia.org/wiki/John_F._Kennedy_document_hoax