OK, what's your application? What kind of environment are you in?
Making some guesses about your situation:
Can you use magnets, vacuum...
What level of thermal drift are you trying to attain?
Thermally matching using similar materials may be an idea. However, even there depending what the heat source is (ambient air?) and where it is applied you may still see significant thermal drift at the scale you're talking about because of the thermal gradient between components.
For instance, you could swap the sst plate for AL. However, if it's SST currently because of corrosion concerns that may mean you anodize the AL plate or similar. This could reduce the thermal conductivity between the 2 parts so depending on the heat source the 2 parts may tend to be at different temperatures so instead of solving your problem you might make it worse!
Could you come at this from the other direction of minimizing changes in temperature? This doesn't necessarily mean a big thermally stabilized enclosure, after all it's temperature of the structure you care about, not the environment as such. Sorry if that's a bit cryptic but this type of question overlaps an area my employer has IP in so I have to be a bit careful.
If you're looking for very low thermal drift with that thermal gradient then I wouldn't be looking at SST or Aluminum without some kind of active temperature management.
There are materials that have low coefficients of thermal expansion that are often used for these types of applications. Invar 36, Super Invar and Zerodur spring to mind. However, you still want to closely match the cte of the 2 components to really minimize the specific effect you're talking about.
Posting guidelines faq731-376 (probably not aimed specifically at you)
What is Engineering anyway: faq1088-1484