At drop of a hat is an idiomatic expression that means to do something immediately or without delay. The opposite of this phrase would be to delay or hesitate before taking action. Other antonyms for this phrase could include being indecisive, taking one's time or requiring a significant amount of consideration before taking any actions. Additionally, reluctance, hesitancy, procrastination, and reservation are all antonyms for "at drop of a hat". When used in a sentence, it can be replaced with "wait and see," "take your time," or "think it over." It's important to understand the meaning of antonyms to fully grasp the opposite meaning of the phrase in question.