She literally changed her outfit to look snowy white for the occasion. She looked like fucking Elsa from Frozen.
Ok, but do me a favor. In a few months when it's late fall going into winter if you live in a place where it snows with some regularity, when the first real snowstorm is coming in, get in your car, get on the highway before it starts snowing, and drive at highway speeds with the window down and your arm hanging outside. Tell me how far you're going to get doing that.
Now tell me when a noreaster comes through how far you're going to get flying at low altitude in let's say a cesna or a Piper cub with the windows open wearing nothing but a London fog trench coat flying from let's say boston mass to Halifax Nova scotia.
Point being, she'd be a meat popsicle before she even cleared the Wall unless she was dressed like a damned spetznaz trooper even if it was a balmy 50 degrees when she left dragonstone.
...my point is that it was very fashionable, not that she changed clothes at all....
But now that you bring it up, it was basically just an inner lining to what the rest of Jon Snow's crew were wearing