Jump the gun
Begin something before preparations for it are complete.
'Jump the gun' derives from track and field races and was preceded in the USA by the phrase 'beat the gun' (or pistol). This has been known from the early 20th century, as in this example from Crowther and Ruhl's Rowing and Track Athletics, 1905:
False starts were rarely penalized, the pistol generally followed immediately on the signal "Get set!" and so shiftless were the starters and officials that "beating the pistol" was one of the tricks which less sportsmanlike runners constantly practised.
The earliest citation that I can find for 'jump the gun' isn't directly related to athletics, but is a figurative usage, as we use the phrase today - from The Iowa Homestead, November 1921:
"Give the pigs a good start; jump the gun, so to speak, and get them on a grain ration before weaning time."
The use of 'jump' in both phrases derives from the 'make a sudden, unexpected movement' meaning of the word. This usage is apparent in earlier phrases 'jump someone's claim' and 'jump ship' and the later (mid 20th century) 'jump the queue'.
See other phrases that were coined in the USA.