(chuckle) Okay, to summarise ...
Yes, once upon a time when there were epaulettes on the shoulders of the old O.D. cbt clo, some people got away with shoving their headdress (usually a beret) into an epaulette (on the shoulder).
However, some messes don't allow that (hear, hear - bravo ... in a stuffy, old accent ... chuckle)
Today, the epaulette (singular) has moved to the chest of the CADPAT uniform, front and centre - I'd be amazed if any PMC in Canada would allow headdress to be stowed there.
In your home mess, you're normally quite safe to leave your headdress in the cloak/coat room (although there are pranksters in every mess).
The further you go from home, the more rare or exotic your headdress becomes ...
Certainly, my headdress was always safe when visiting the Scots Guards or Gordon Highlanders, but ... there were some other "more seedy" messes ... where I'd fold my balmoral up and secure it in the cargo pocket of my cbt pants.
Alternatively, the glengarry can be secured in the inner breast pocket of your DEU jacket (offset by your wallet in your opposite inner breast pocket, giving your that "enhanced" chest profile you've always longed for)
And, if somebody really wants to rip you off, they'll find a way (like the time they reached through an open window in the old 'H' huts in Borden ... and stole my uncles' Second World War cap badge from me ... the rotters ...).
Now, having said and read all of the above ... it shouldn't be such a mystery. Good manners dictate you remove your headdress indoors, unless on duty (as an aside, when working with American forces we've discovered they ALWAYS remove their headdress indoors - so, to be polite, we followed suite so as not to appear gauche, but ... I digress ...).
There's one final reason why a person wouldn't really want to take their headdress inside the mess and leave it lying around ...
Your PMC or others might decide to teach you a lesson (the hard way) - I'll now let your imaginations run wild ...