Oddly specific question, yes, but there seems to be some information available on the history of undergarments in Europe as a whole - yet no source I've come across has mentioned anything about what the eastern nomad people wore under their furs. Any insight on the matter is welcome, thank you in advance.
Ammianus Marcellinus has a bit about the clothing of the Huns, which you may or may not find useful.
The Roman History, 31.2.5-6
They dress in linen cloth or in the skins of field-mice sewn together, and they wear the same clothing indoors and out. But when they have once put their necks into a faded tunic, it is not taken off or changed until by long wear and tear it has been reduced to rags and fallen from them bit by bit. They cover their heads with round caps and protect their hairy legs with goatskins; their shoes are formed upon no lasts, and so prevent their walking with free step.
While this does not specifically answer the question of undergarments, it at least gives some of an idea about the kinds of textiles the Huns would have used: furs, as you've mentioned, but also linen. Given that linens have traditionally been a common undergarment material in the West, it seems quite possible that the Huns may have worn a type of linen loincloth.
It's also worth mentioning that Ammianus Marcellinus (or any other Roman source, really) cannot be considered a highly objective source of information about nomadic steppe culture. There is a strong element of polemic in his treatment of the Huns, which may be distorting the information here.
What does 'their shoes are formed upon no lasts, and so prevent their walking with free step.' mean?