People wore coats to dancehalls in those days. There were cloakrooms and they gave you a little ticket, but if you did not get your own coat at the end of the night there was no real comeback, no liability. Then there were places that did not have cloakroom attendants, That was just asking for trouble.
'The Big Red Building in Petticoat Lane' was a memorable radio jingle for a famous sheepskin outlet. Would have been early 70s though - so past the boom era.