So what's the deal with a push dagger, (also known as a 'fist knife' and a 'punch dagger') and why would a gunfighter carry one?...
It's a "T-handled" knife, typically with about a 2-inch blade, and is held in the fist with the blade protruding from between the fingers. A push dagger is a fighting man's knife, a last-ditch, close-in, down and dirty weapon, and with it, you can slash and punch your way out of just about anything. (Short of bringing a knife to a gunfight.) They were popular with brawlers and toughs and card sharps of the 19th century, sometimes found up a sleeve, sometimes slipped into a boot. Push daggers were used in the trenches by soldiers of both sides in WWI, and were issued to the British Commandos in WWII. It is my understanding that they are still in use by the SAS.
Push dagger trivia: The Montagnard tribesmen in the central highlands of Vietnam were clever at making their own little push daggers... They would take an ordinary icepick, remove the handle, and re-fasten it horizontally into a T-handle. Their favorite thing to do was to slip up behind a man, cup his mouth, and drive the icepick straight into his ear.
In your pocket, on your belt, in your boot, or up your sleeve... Another real good close-quarter backup weapon, and in .41 caliber, a man-stopper.
And just so you know I know... "derringer" is wrong. It's a misspelling -- albeit one in common use -- of the last name of Henry Deringer, the man who made the little pistol used by John Wilkes Booth.