You can probably push padding back as far as the first metal helmets — so yes, into the Sumerian period.
The copper helmets from the Royal Cemetery of Ur (e.g. the one associated with Meskalamdug, c. 2500 BC) are thin, close-fitting, and would transmit shock very efficiently. Wearing one directly on the skull without anything underneath would be, frankly, a bad idea. So metal helmets & padded lining would have always been together.
We don't have surviving liners (not surprising — textile/leather almost never survives) and no written reference saying, "they wore padding." But there are a few clues.
Edge treatments and fit suggest the helmet sat over something
Contemporary depictions like the Standard of Ur show snug helmets, possibly over caps
Every later culture that uses metal helmets also uses padding of some kind
So the sensible interpretation is that even in the Early Bronze Age they were wearing simple caps underneath — wool, linen, or felt would all do the job.
The real difference isn't whether they used padding, but how much:
Bronze Age: probably thin cap or basic liner
Roman: internal liner (leather/textile)
Medieval: heavy, purpose-built arming caps + more developed suspension
There's no reason to think anyone at any period happily wore bare metal on their head if they could avoid it — we just have to infer the early solutions because the materials didn't survive.