'paper tumbler' is actually used with that meaning. (I'm not saying it's common, though). A Google Books or Google search will find more examples.
Song of the Atman, by Ronnie Govender, 2006
One of the minstrels thrust a paper tumbler into his hands and filled it to the brim from ajar of Oom Tas wine - which he would later discover was referred to as "rot-gut".
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