Very interesting! Those red/black probes use high-quality pomona grabbers on one end, but the banana jack ends are just cheap ebay plugs.
In the above graph where there is a quiet section in the middle, was that when you put the reference inside of the tin? Or was that when you left the room?
Have you done a comparison of inside / outside the tin?
On my end, I finally managed to get a more reasonable result with my Keithley 196 and a few LM399's I've been playing around with. Previously, if I took three steps to the side, the value would climb by about 40 counts, and then come back down if I moved to stand directly in front of the meter again.
I noticed that the reading jumped if I connected an oscilloscope lead to the board (in fact, just connecting the ground pin changed the value), so I suspected an EMI problem. However, putting the board inside an aluminum caldero didn't seem to change this 40 count drift at all.
I was surprised to find that simply putting a 0.1uF ceramic cap directly across the (zener) output of the board has reduced that drift to 1 or 2 counts, even when it isn't shielded at all. (I was surprised because I don't see such a cap in e.g. the Keithley 196 reference schematic!). That change (combined with removing the lid of my 196, which seems to be running hot), has the output of this board staying within 1 or 2 counts all day today (from a zeroed point this morning).