Thanks all for the thoughts. Air flow was not the problem. The computer does not look all that nice, but there is a decent fan in the front behind that plastic bracket at the lower right of the picture. I did remove a couple of rather minor dust cakes inside the card, but there really was not much to speak of. I don't think it was restricting air.
I did reapply what little thermal grease I had on hand though. I only had enough for a super thin layer all around. I reused the the thermal pads. Either the old grease dried out (probably) or just the act of disassembling and reassembling the card allowed for better thermal contact. I am going to disassemble and do it again the right way once I get a new tube of paste and some padding.
It is running much quieter now.
That nzxt product is exactly the kind of thing I was thinking on. It looks like plenty have modded it slightly to work with 3000 series nvidia cards, and it is relatively inexpensive.
I actually managed to quiet down my card by essentially replacing the thermal paste. I did cut away some metal on the case that acts as a divider between the two card slots, increasing air flow. I am not sure which of these contributed more to the overall success, and while I can still technically hear the fan when pushing the card, it is hardly noticeable, even with no audio playing. I think I am happy with it.
That said, I am glad I created this thread. There are lots of good product recommendations, and I may move on to water cooling at some point.