Baeyeaux Tapestry shos something, and I know that the Romans used padding in their Hamata.
Yes I also noticed that. Perhaps just a shirt.
Perhaps The possibility of just somewhat thick shirts before late 12th century and as time went on it began to evolve more and more, to what we know now, the fact that it was used and differentiated as the middle ages went on.
However, early great helms pop up in the early 13th century from multiple regions and sources, not necessarily just the second half.
True, what I should have said was that later style great-helms appear in the second half of the 13th. The one in the armoring sequence looks like a later, barrel-shaped great helm to me, rather than the high-backed, or faceplate styles that you tend to see in the first half of the century.
Those Helms look like topfhelms, so Faceplate/ Highback styles, but those were a lot less protection than the helmets we begin seeing in the mid to late 13th century. I think Faceplate style started in the late 12th, although it definitely was not the Norm of helmets to have back then.
the picture should be called 13th century. Since evidence of gambesons closer to that exists more, and the type of helmet we see in the sequence.
From a practical stand point, mail without padding is not good protection against swords; you won't get cut, but bones will break like candy. And broken rings driven into flesh, from not having padding, would be an unpleasant experience (if not death from infection).
I have no problem wearing mail over a t-shirt to walk around or piddle. But for combat? No way. I'm also a soft 21st century guy, so, who knows. 
Well Even though mail and padding seem to go hand and hand, It is quite possible that just a thicker shirt was worn. Since people still used shields as their main defense, the mail was only there to help against cuts. Because even if you have thick padding, If someone hits you with a mace or sword, It is Going To hurt Very bad.
(since we know due to the start plate smaller shields were being used by late 13th, but padding was still essential)
Look at this image that dates to 13th century. Their are no shields Perhaps because as armor got better(during this era would be mail and helmets, then plate.. but that was later) there would have to be a reason none of these men are using shields(perhaps because of thicker padding, or maybe it was a test of skill.) I also remember reading that there may have been a type of mail for jousting that was extra thick.
But what this image would show is that armor was protective enough that the combatants would be willing to not use shields. Where as in earlier century's the shield was the most important protection. more padding and better armor could have been adapted as people needed to be better protected if they lost a shield in combat. but in earlier century's armor protection was less adapted.
I guess what I make of it is that padding evolved just as armor did. Weather or not this is already a known fact, I do not know.
