Sorry, it doesn't look like your graph is showing anything, really, with due respect. First, if ambient temperature went up 7C, from 21 to 28 before AC on, it should have very similar effect on TEMP? output of the meter too. But that's not as issue, as reference output. It's span 7ppm, which is not what LTZ-based module should do, even on much largest temp span. So the setup is compromised and any correlation of temperature to output is just buried down down under the interference to LTZ module (be it EMI/RFI or air drafts or mechanical stress to cables or module).
Suggestions to improve:
* I don't know what logging sw you use, but add ACAL DCV function on each 0.5C of TEMP? change. This will remove tempco from your 3458A, down to <0.05ppm/K. My 3458A has tempco on 10V DCV range about 0.35ppm/K without ACAL, to give you the idea.
* Use twisted pair short copper wire between your DUT LTZ and 3458A. Put foam around binding posts to isolate from air drafts. You can use regular LAN UTP cable, it works great for this case. No need shmick 100$/meter teflon silver-99%-of-light phoolery here, just pure fresh cut copper will do great.
* Make sure your PSU is quiet. You can test it with 3458A , just log bit of data to see pk-pk. SMU should be fine, I use my 2400 all the time for LTZ tests, but I checked it before comparing with battery powered ref, to ensure setup is good.
* Turn off all PWM-dimmed lights in the room, those LED ones and such. Often the radiate into mains and that get coupled.
* Air drafts. Your biggest enemy with LTZ. Your task to have as still air possible. It also meaning filling with foam, just putting ref in the box not enough, as thermal gradients cause air to move.
* AC cooled room can oscillate. Meaning that your AC will be turning on and off to keep temperature setpoint, and that will cause temperature oscillation. I have it happening if I close the door in the room with gear and AC on. So I always leave door open and have AC only in the gear room, so hot air from outside dampen the temperature changes. By doing this I can reduce oscillation from ~1c down to ~0.2c. And it's no basement stuff (envy people with houses), just subtropical apartment building with thin walls and regular split HVAC unit.