At the drop of a hat means immediately, and without needing to be asked twice. He would do it at the drop of a hat, cries at the drop of a hat, travel at the drop of a hat.
It describes eagerness rather than speed, and it is often faintly critical of somebody too ready to act. The image is a race started by sweeping a hat downwards.
1phraseat the drop of a hatImmediately, and without needing to be asked twice.
“He'd fly to Paris at the drop of a hat if someone asked him.”
at the drop of a hat is a readiness.
From starting a race or a fight by sweeping a hat downwards as the signal.
From the customary starting of competitions by making a sweeping downward motion with a cap.
Etymology from Wiktionary, CC BY-SA
Practised in Idioms