As Jock and Matt have said, the flat cap issue referred to is exclusively an XMTS one and was got up and going as a way to take the michael out of a certain Lochaber-man.

The baseball cap has now been reduced to "ballcap" has it? May I suggest that that moves it somewhat closer to being accepted as tradtional US attire I'm not that familiar with its history, but it seems to me that the primary intent to keep the sun from one's eyes and less important purposes related to warmth and dryness indicates its origine in the summer sun or the southern part of your country.

The utility of the flat cap is the reverse: warmth and dryness first, eye protection somewhat after, and I suspect that it had its origin in England among the labouring classes, migrating north into Scotland with the industrial revolution. Certainly its fashionable wearing by the landed classes in either country (and Wales and Ireland) was very, very brief and gone with that class's reduction by 1918 or so.

The issue is not whether the flat cap is traditional to Highland dress, therefore, because it is not. Nor is it current fashion with Highland dress in the Highlands today. It is periodically seen, but at some sort of gut level we just know it to be a no-no at worst and a private style thing at best. I suppose a bit like wearing a ballcap with a suit.

As Jock says, the deerstalker and fore-and-aft are different kettles. They have been worn with Highland dress for long, long years and may just be approaching the "welcome to tradition" level of acknowledgement. Mind you, we know where and when they are appropriate, too.

Rex