Yes, to a degree. You’ll need a hat jack/stretcher, a source of steam and some patience. You’ll be elongating the hat, so you’ll want to move slowly to protect that reed and to keep from distorting the brim. Only use steam on the outside of the hat, keeping it away from the leather sweat.
Sorry no one answered this: in the US we just had a major holiday.
If it looks good and feels like you can live with it. Go on. If it hurts or visibly looks 'off', you can make a judgment call.
Do you know the shape of your head? You are not really stretching the hat if it is the right size for you just the wrong shape. If you know the shape you can cut yourself a band block out of 1" plank or plywood. Then when not wearing it slip in the band block and it will help conform the hat's shape to your head. I am a long oval and have a few stiff western weight hats. I use the band blocks to good effect to turn their regular oval into a long oval. If you don't want to cut the band block then a hat stretcher can replicate one just be cautious to not use it to stretch the hat.
John Lofgren Monkey Boots Shinki Horsebuttt - $1,136 The classic monkey boot silhouette in an incredibly rich Shinki russet horse leather. 
Grant Stone Diesel Boot Dark Olive Chromexcel - $395 Goodyear welted, Horween Chromexcel, classic good looks. 
Schott 568 Vandals Jacket - $1,250 The classic Perfecto motorcycle jacket, in a very special limited-edition Schott double rider style. This sounds like a really good idea. How would I best go about transferring my head shape onto paper?
Go to Amazon and buy a Flex Curve, 36" (if your head is smaller than 7 1/4" you can get away with a 24". Get someone to wrap the Flex Curve around your head making sure to get a tight fit to the head (No gaps!). Gently slip it off your head and voila, you have your head shape. This achieves the same result that you would get by using an expensive conformateur. Use this as your template to cut the band block. Drilling 3 large holes for your fingers to ease placement in the hat.
Fantastic idea, thank you!
They dont look that bad to me. I'm used water stains on my boots in the snowy seasons... adds character. Items on the ground tend to wear out and get weathered more rapidly. If they were mine I might take a damp rag after them and polish a little bit later, but that's a risk u may not take.
agreed, they are not terrible and I can definitely live with them. Ideally I would get rid of them.
Reminds me of Rat Pack Era early '60s.
Beautiful hat—I would guess from the 1950s. I would not try and reshape this hat—I think you would only ruin it.
I would leave it as is, a true classic!
