Hi Erik,I think you need to be looking at South German and Milanese examples, as that seems to be where the Swiss got their armour from principly. I don't think they had a large manufacturing capability locally.
If you look at Schillings chronicle of Berne, the infantry of both the Swiss and Burgundians are illustrated primarily in kettlehats, but open ones, on the order of deep doughboy helmets, or old CD helmets - I am sure they had great variety, but the illustrator supposedly was a participant (as was the author) in the wars, so they have some credibility for probably depicting the commonplace.
I think Peder has some of the illustrations scanned and online, under his investigation into Haversacks and canteens. If you look for that topic here, you will find a link to his website.
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Bob R.