The stability / noise looks good. For the noise it is hard to tell how much comes from the meter and how much from the reference. So even the low current is sufficient for a low noise reference.
From the long run, it looks like much of the variations is still due to temperature variations. As the current is already adjusted for a small TC, one might need temperature stabilization to get a considerably more stable ref.. Current adjustment can only compensate the linear TC, but the second order effect still remains. Here temperature stabilization is likely more promising than even more adjustment / compensation.
Looking at the two points, where the diode shows 717 mV, there is a small but visible difference (about 2 ppm). This is more than the expected uncertainty of the meter. So this would be a first indication for aging or maybe hysteresis, the two still unknown parameters.
A phase of stable temperature gives a nice looking curve, but not much to learn from this. A similar measurement with more temperature variations would be more interesting.