The word existed before its slang usage. Originally, gumshoe referred to a shoe with a rubber sole. Though it might refer to rain boots or galoshes (more on that later), in this context it refers to what we would call today sneakers. A gumshoe would be a private detective in the sense that a rubber-soled shoe would give the wearer the ability to walk stealthily.
At the turn of the century 20th century, "to gumshoe" meant to sneak around quietly as if wearing gumshoes. It could be to rob or, conversely, to catch thieves.
Apparently, a "gumshoe man" was originally slang for a thief, but then around 1908 "gumshoe" seems to have meant a detective, as it has ever since. Some dictionaries list uses from 1908 as gum, gumfoot, gumboot, gumheel, gumshoe artist and gumshoer as also meaning a private detective.
That other gumshoe footwear, galoshes, is a word that comes through French and Latin from Greek and originally meant a shoemaker's last. This was the literal "wood" + "foot" that a shoemaker used to form the leather around as a form.
The original galoshes are not very similar to what you would be shown if you asked for them in a store today. They were once English style clogs with a wooden sole and fabric/leather) upper. By 1572, the term also applied to "a Gallage or Patten" which was an overshoe with a shaped wooden base to raise the wearer's good shoes off the ground in wet weather. "Goloshes" appears to be the older spelling of galoshes used previously in Great Britain.
In modern usage, galoshes are outer shoes worn in inclement weather to protect the inner shoes and keep the feet dry and are almost universally made of rubber.
|Some fashionable galoshes - via sharperliving.co.uk|