At the drop of a hat means at the slightest provocation or without any delay
Her baby cries at the drop of a hat.
They are ready to fight at the drop of a hat.

One waves hat in greeting or to express enthusiasm, throws in the air to mark a victory, raises to acknowledge a passing acquaintance, or drops to send a signal. This last action has been taken to be what at the drop of a hat means today.
In the 19th century it was the practice in the United States to signal the start of a fight or a race by dropping a hat. The immediate response to the signal finally saw its way into the language for any action that begins quickly without much prompting.
However, it is usually assumed that it indicated that a formal fight was to begin.
The editions of Brewer’s Dictionary of Phrase and Fable comment that “The expression alludes to the American frontier practice of dropping a hat as a signal for a fight to begin, usually the only formality observed.”