EDIT: Changed board design to a simpler LM317 constant current source.
Welcome to the path of becoming a volt nut!
The last time I fooled around with the LM399 and read through the LM399 mega thread on this forum, if I recall correctly the stability of the LM399 voltage ended up being dominated by your ability to drive a stable current through it. I can't remember the exact relationship off the top of my head, but if your current source drifts by X microamps, the 399 drifts by Y microvolts. (I believe this means the LM317 is going to be the biggest source of error for your design).
The design presented in the LM399 mega thread (
https://www.eevblog.com/forum/projects/lm399-based-10-v-reference/ ) is a really interesting one, because it uses the LM399 itself to control how much current goes through it. Its kind of hard to wrap your head around, so I drew up a simplified version of the schematic in a way which makes sense to me:
That circuit produces roughly 10 volts at the output, but you can simply change the two 3k resistors to whatever you want to get a different output voltage (e.g. 4k would give you 11 volts, 5k would give you 12 volts...).
The key thing to understand is that, as soon as there is any current at all going through the LM399, it is near 7V. That causes the op amp to try to make its other input close to 7V, and the only way it can do that is to drive 1mA through the 7k resistor. And once that happens, you are also driving (a very stable) 1mA through the LM399 as well. Pretty neat!
Here's an idea I've been sitting on for a while (which is based on the above circuit) to create a mini transfer standard to calibrate the various ranges on my HP 3478A bench meter: