The "Hat" thing is bizaar.
Etiquette dictates that
1) Gentlemen should not wear hats indoors.
2) Gentlemen should not wear hats in vehicles.
and
3)Common sense dictates that you don't wear a hat on the apron as it's a potential FOD hazard.
So taking into account 1+2+3 when would be the time to wear said head garment?
Just asking