1
general::
Immediately, without delay, as in We were ready to pack our bags and go on vacation at the drop of a hat. This phrase probably alludes to signaling the start of a race or other contest by dropping a hat. [Late 1800s]
American Heritage Idioms
2
general::
Phrase(s): at the drop of a hat
Fig. immediately; instantly; on the slightest signal or urging. (Alludes to the dropping of a hat as a signal.) • John was always ready to go fishing at the drop of a hat. • If you need help, just call on me. I can come at the drop of a hat.
McGrawhill's American Idioms And Phrasal Verbs