Shoes in the late 1800s were made of gum rubber - the soft-soled precursors of the modern sneaker. The phrase "to gumshoe" meant to sneak around quietly as if wearing gumshoes.
An old slang term for a detective or investigator (police-affiliated or private).
By about 1908 the term "gumshoe" was used to refer to a police detective, and the term has stuck ever since.
Really adds some meaning to gun shoe.