Again, there is confusion over terms.
There are two kinds of padded garments;
A padded garment worn under plate or mail, often reffered to as an Arming Coat
and the independent cloth armour; Acceton, Jack, gambeson... I don't speak medevial french, so I really can't tell what is more correct. Its called Panzar in medevial norwegian...

A picture of cloth armour, in the macieowski bible, ca 1250 (http://www.medievaltymes.com/courtyard/maciejowski_images_27.htm):
[ Linked Image ]

A regular man-at-arms would be more likely to wear cloth armour than mail, mainly for reasons of cost. Mail is however predominant in the illustrations, because it's cooler. :cool:
Both will do, of course. But you should in any case have a arming coat under the mail...

So here's an oddball question.

All this wool, how do you keep from scratching yourself to death?

Or, you just make sure to have finely woven wool. The wool you use for for instance hose is comparable to the one used to make suits. Rough linenen actually itches more, so be careful to get a reasonably soft fabric...
Personally, I never had problems with itching. But it can get a bit hot after a while. ;)

I do 15th C. and have worn woolen hosen in all kinds of weather hot and cold without discomfort. When I first started this game I thought wool would be terrible, especially since I don't od heat very well. Properly constructed period clothing is no worse in hot weather than modern clothes. Elling is right, it has to be a finely woven wool if it is going to be worn next to the skin with any degree of comfort. Be sure you pre shrink it if you are going to try and do it yourself. Google medieval hose and search through the sites. There is plenty there that will help you.