To do something at the drop of a hat means to do it immediately, without delay and at the slightest provocation. The idiom may have come from the American Old West, when various fights, contests and duels began with a signal consisting of a man grabbing his hat and thrusting it toward the ground. Alternatively, the idiom at the drop of a hat may be of Irish origin, stemming from the practice of signalling the beginning of a fight by thrusting a hat in a downward swoop. Today, at the drop of a hat often carries the connotation of an overreaction, of being too eager to do something.
The views of this African-American pit master suggest that many Americans are curious about Trump not just because he brazenly breaks the rules of political rhetoric but also because the key to tackling America’s problems, they say, requires people to stop taking offense at the drop of a hat. (The Christian Science Monitor)
They also may be called out at the drop of a hat to distract the crowd if there ever is a problem or safety issue with another act. (The Washington Times)
“He would get into fights at the drop of a hat… [he had] swallowed whole attitudes and behaviours of masculinity they’ve witnessed growing up and at this point in his life, he hadn’t really done anything to change those.” (The Huffington Post)
“Connie cries and screams at the drop of a hat as a result of her daughter and son who were killed. She can’t get over it.” (The Johannesburg Sunday World)