G'day Tdol,
I checked the link you supplied and found the definition given to be a little simplistic, if I am allowed to say that here.
I mean no offence but to say that, 'At the drop of a hat' means simply 'immediately' or 'rapidly' misses the entire point of the saying.
At the drop of a hat can mean instantly but it can also mean that a thing will happen instantly but at some time in the future.
The full idiomatic saying is similar to, 'He will fight at the drop of a hat!'
What this means is not that he will fight instantly or even soon. It does not mean that he will actually fight unless someone 'drops a hat' but blokes are very careful about 'dropping hats' around him.
To drop a hat is one of the quietest and most innofensive slow gestures that a man makes.
The hat falls slowly.
The hat makes no noise in the falling or the landing.
The hat damages nothing.
A dropped hat is not thrown.
There is nothing offensive about dropping a hat.
A bloke who will blue at the drop of a hat is a bloke who will fight at the slightest provocation or excuse.
To do something at the drop of a hat is to do something on a whim.
The whim may be instantaneous or delayed but it is the whimsey aspect that the saying is trying to convey.
See ya
.,,