To do something at the drop of a hat means to do it instantly, without any delay.
If you need any help, just phone me. I'll be there at the drop of a hat.

The idiom may have come from the American Old West; in the 19th century it was occasionally the practice to signal the start of a fight or a race by dropping a hat or sweeping it downward while holding it in the hand.
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Asynchronous learning At law At the drop of a hat attend Attorney