Sir Nathan, a linen surcoat, preferably lined, is the way to go. My hauberk's riveted and all of my surcoats are either cotton over/linen inner or linen/linen. The inner lining helps protect the garment from the stresses of being worn over armor.
I still need to get a custom one done, either by BJ or Brian's seamstress...but not until I'm sure that my CoA is one I want to go with moving forward.