I don't understand why people folded up one side of their hat's like that. I've seen it done with other hats, too. I could understand if they did just the front (so it wouldn't obscure their front vision) but why just one side?
One of the upsides to being unemployed is it gives you time to research things you normally wouldn't research. Here is what I found out about the origins of turned up hat brims.
The first manufactured hats were simply a round piece of leather. A ring of holes about the size of a head was punched in the leather circle. A string was woven through those holes. The string was then pulled tight to hold the hat snugly to the head. The excess material created the brim.
To keep the wide floppy brim out of the wearer's eyes it was tied up with a ribbon. It was later discovered that sometimes when the ribbon was removed from the brim it would stay curled by itself. This led to hand curled brims.
To keep them from fraying hat brims were bound with ribbon. Although advancements in materials have eliminated the need for binding the brim, or pinning it up, the custom remains. To this day we keep our brims curled up because early hatters did not know how to stiffen a brim.
Here are a couple of additional bits of Hat Trivia: When men went off to do battle it was customary to wear a feather from their loved one in the adjusting hatband. Since they were mostly right handed, when sword fighting they lead with their right side. To keep from having their view obstructed at this crucial time they took to sticking the feather and the knot of the adjusting hatband on the left side of the hat. As time went on, they would wrap a ribbon around the crown to hold the feather and hide the tie string knot. Where they remain today.
When the material used for hats turned from leather to velvet some protection was needed to keep the softer velvet fabric from falling on people’s hair, this is where the lining came from. Even though modern hats are stiff enough not to collapse, the custom remains.
Individual sizing eliminated the need for the adjusting tie string, but the bow remains at the back of the hat, serving as a memorial to bygone hatters.