So, apparently the title is a (dated) saying much like, “The devil always gets his due.” That is to say, stuff comes around & if it doesn't get you the first time, rest assured, it'll get you eventually. So that scene with an actual mailman? Yeah, it's not in here. This book, in fact, has a lot of language (the title aside) now discarded. The other striking expression used quite frequently here is the verb to "tumble." It is used much like we would use the verb to stumble onto. Characters would hatch a plan & then would chicken out for fear that a cop would tumble (i.e., get wise to their plan).
Other things I don't know about this book going into it (other than that there is no mailman character in it)... I don’t know this is a noir genre crime story! Which is to say, I know nothing about this book starting out. I remember the 1946 film not at all & was just at an age when I was cognizant of the 1981 film (with Nicholson & Jessica Lange)—and that steamy lobby card for it—but was forbidden to take a peek at it any one of the 789 times it ran on cable. From my mother's caveat alone, I knew it was a "love" story...and that's about it.
Postman is a little book; a thin novella, vividly executed. James Cain's style is one now oft imitated, but apparently he was one of the first to write in this manner: gritty, realistic, raw, dark, unapologetic. The first-person narration is tight & engenders a reader's sympathy for the narrator (who would otherwise simply be a crummy person). This is a crime drama, folks; complete with ignorant people executing gore and even a courtroom scene or two. A classic!