My primary care doctor ordered a morning cortisol level when I wasn't feeling well on 3 mg of prednisone. It is a routine lab that should be done if adrenal insufficiency is suspected. When my cortisol level was low, I was referred to an endocrinologist.

"Generally, a random serum cortisol of over 400 nmol/L at any time of the day makes adrenal insufficiency highly unlikely,8 while a morning serum cortisol of less than 100 nmol/L strongly suggests adrenal failure.13 In interpreting such results, one must consider the patient's current and prior steroid usage."
https://pmc.ncbi.nlm.nih.gov/articles/PMC6297573/
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There is a fancy cortisol level test that is called a synacthen test that is more involved and probably requires an endocrinologist to interpret.
https://pubmed.ncbi.nlm.nih.gov/29846658/
My endocrinologist didn't think I needed a Synacthen test. She expected my cortisol level to be low given I was on Prednisone for a very long time.

While my a.m cortisol level was low, my endocrinologist was encouraged that it wasn't dangerously low. I was told to stay on low dose Prednisone (less than 3 mg) for an extended period of time. After 6 months my cortisol level improved and it was deemed to be "adequate."