^^ I own the ambient.
Yes, very small brim, not anywhere near as big as a regular cycling cap brim. The brim is maybe 1", if that, and one of the reasons why I purchased. The fit is snug, not loose at all and it covers my ears which many caps do not. I have not used it in the cold yet but compared to the dozen or so other hats and caps I own, I have a feeling that this is will be one of the better ones.
The OP is asking about thin caps. This is not very thin. It is for cold weather, not cool.
I also own the Castelli Extremo Windstop Skully. It is similar but without a brim and less ear coverage for those with big ears. On sale for $20 at Excel.
Fixed Link -
https://www.excelsports.com/main.asp...ajor=4&minor=3
A nice thin cap is the Giro Merino Winter Cap. Very light and soft. It is a classic looking Mutt, AKA Belgian Cap, great for cool weather but has a larger brim.
https://www.backcountry.com/giro-merino-winter-cap
-Tim-