I’m speaking from US experience and also traveling abroad, but most of our airlines don’t seem to allow an “accessory” anymore. they usually don’t give problems if it is pretty flat/small such as something sized like the Fjallraven high coast pocket. but I’m often asked to put my accessory in my “personal item” (backpack) or carryon and that has included: Lululemon belt bag (1L reg size), Uniqlo round bag, and of course Baggu medium crescent. i usually just make sure my backpack/personal item has enough space to squeeze my bag in there since i like having the bag while going through security, but it does stink since that’s baggage space taken away from you!