I am reading The Maltese Falcon. Though the book was written in the 1920s, the edition I'm reading was printed in 1992. The back cover blurb uses the phrase "at the drop of a dime" to describe the sudden changes of loyalties in the story.
Some sources note this phrase's similarity (in meaning and construction) to "at the drop of a hat". Though I can find good explanations for Why hat? (most centering around the hat being used like a flag to signal the beginning of a fight), I cannot find anything similar for Why dime?
It may be a malaphor, a neologism that means
An idiom blend: an error in which two similar figures of speech are merged, producing a nonsensical result.
As one blogger says:
[It seems] like a combination of “do at the drop of a hat” and “he dropped the dime”.  Very different meanings, but the word “drop” apparently led the speaker to think “dime” instead of “hat” (alliteration perhaps?) and thus another malaphor was born.
Given the popularity differences displayed in this nGram it would make sense for at the drop of a dime to be a corruption of at the drop of a hat.
Are there formal sources on the origin of, or use of dime in, the phrase at the drop of a dime?
 The phrase he dropped the dime may mean to make a telephone call, particularly to snitch on