The only way to raise the strength of an 300 SS is through cold work.
The strain hardening will make it stronger, and a little less ductile.
This material will also have a lower service temp as high temps will result in stress relief.
Why do you need the strength?
Why do you need 316?
What about Nitronic 50, or a duplex SS (LDX2101)?
I will add, the material strengths are defined in the material specification, not by the alloy name.
What product form are you working with?
Do you know which ASTM specs cover that form?