I think if you go regimental in a kilt you are supposed to wear an extra long shirt, like almost to the knee. Not sure, but that's what I would do, so when you take the Kilt off at night to use it as a blanket for sleeping under you are basically wearing a night shirt. The best material for such a shirt becomes an issue. I would say wool again, but light in weight and in colour. I think in ancient times these were dyed saffron. I would think they would be made of linen, from flax of course. I would think that linen would suffer similar consequences to cotton when wet. Not sure. Moot point if unavailable.
My choice for a shirt when hiking with a kilt would be a shirt made of a blend of cashmere and silk, dyed saffron and cut long to just above the knee. Might not be easy to find such material.