I have used a small backpack on my chair (though the one I have isn't great as it's meant for a laptop and has far too much internal padding for general luggage use), but I'm actully thinking about a full size frame rucksack, which is probably long enough it'd need to be hung from me rather than the chair, I have one I can try out, but I may look into a better one (and I think there are a couple of shops in town that should have a reasonable selection)

Underchair bags are unfortunately out as the monster is the under seat cross-braced type.