What makes you think the mail does not have padding beneath it?
Mail without padding leaves you removing metal rings out of your flesh, and chaffs worse then sandpaper.
Full plate, and half plate both always have padding underneath them.
Scale Mail as well.
As the proud owner of a hand made chain mail coif, you always wear padding under your mail. Always.
Finally, there was a period where plate over mail and padding existed, a brief period of time, then they got smart and relized that any blow that could kill someone wearing plate killed then regardless of the chain mail. (Not counting the protecting the chainmail provided to the joints)
Because weapons kill people in plate harness via concusive force, not by piercing thru the plate ( unless you have a 800+ pounds horse giving power to your lance).
They then just made small pieces of Chainmail sewn to the padding at critical points.
And the padding worn under plate harness is not as thick as a gamberson, as that would have drastically restricted mobility, increased weight, and made those wearing it die from heat exaustion during winter (minor exageration).
And from a mechanical standpoint of 5e.
5e relies on bounded accuracy. Anything that gives an extra point of ac here or there is potentially balance breaking, a magic +1 chain mail coats 1000s of gp.
Anything that gives bonuses should be looked at with extreme caution.
So not only should your chainmail already have padding, though not as thick as a gambisen, it would make the game less challenging, ergo less fun.
You are adding a bonus by spending a couple of gp, a bonus that should cost 100 times that, and might not ever happen (DM fiat).