Okay, so, as far as our sourcing goes, the answer is 'probably but we only have a 13th century text to suggest that they did' which is really the answer to a lot of viking-age Scandinavian history.

The family sagas (which deal with the viking age) are all written several hundred years after the fact, and concern themselves less with the practical day-to-day of life in early medieval Scandinavia, and more with the politicking and adventuring of a few wealthy and famous Icelanders.

As far as actual evidence goes, most of the saga heroes who wear armour do so on the continent or in England, and do so in the company of professional warriors, so it would be reasonable to assume they're wearing some sort of padding below their armour. That said, though, the padding itself doesn't need to be particularly thick as it would also be rather uncomfortable to wear. Likely, mail was worn over a thicker overgarment worn over regular clothing, but it wasn't padded armour by any stretch.

Konungs skuggsjá mentions, in Ch. 37, that a gambeson of soft linen is the chief armour onboard ships and should also be worn beneath mail, but given that it's described as 'soft linen,' rather than 'firm' (which is a differentiation made later) it's not likely that it was a particularly dense garment.

This is a website run by Viking age interpreters- I've found that interpreters often have an incredible combination of scholarly expertise and practical experience because they actually make, wear, and work in the clothes to recreate the experience of people in the past. There aren't many sources here, but it's an excellent primer on terms and concepts if you'd like to look deeper.

http://www.hurstwic.org/history/articles/daily_living/text/clothing.htm

This is another source for interpreters and reenactors (there are subtle differences between the two, mostly related to the degree of study and dedication to accurate experience), and this one provides contemporary source citations:

http://www.vikingage.org/wiki/index.php?title=Jacks_and_Gambesons

It is probably reasonable to assume, even given the lack of period sources, that the under-mail garment was similar among the Norse as in other areas of medieval Europe- a shirt-like creation with at least two layers, possibly quilted and/or stuffed with a buffer material like wool, and possibly with a leather outer layer to provide waterproofing or additional layer of protection under the mail. Over the undergarment and mail, it was common to wear an overshirt like a Norse kyrtle. Kyrtles, surcoats, jupons, and tabards are all related garments with similar functions.