Below is a School Blazer Badge and Hat Band with Badge, dating from around 1937.
The hat band was worn by the girls in summer on a straw hat and in winter on a velour one.

They were loaned by Daphne Watson (Elliott 1938) and Polished and Scanned by Edmund Hodges. 

Close up of the above hat band badge.

Patrick Williams, kindly sent us this picture from around 1943 showing the hat band. You can also see him in a school cap from that era.