Hatband n. : A band round the crown of a hat; sometimes, a band of black cloth, crape, etc., worn as a badge of mourning.

Example:
Where does the expression "tighter than Dick's Hatband" come from?

Taxonomy
crown, brim, hats, military, felt hat, Hat size, hatband, brim hat, puggaree, free encyclopedia, visor, fur, fur hat, peak, sweatband, band, Straw hat, the crown.

Defination from the web:
hatband (American) – a ribbon or band that runs around the bottom of the torso of the hat. The sweatband may be adjustable with a cord at the top and is on the inside of the hat touching the skin while the hatband and puggaree are around the outside. ...