These things are unique to you. Only you will be able to find these answers. Read about thyroid meds to get some sense about what you feel in terms of when it might be too much for you.
T3 is fast acting. Is your T4 level considered optimal now? If so, you should explore:
If you have trouble sleeping, shift afternoon dose forward or reduce.
If T3 helps you, there will be initial effects and some longer term effects that can change everything else.
You should be taking DHEA and some progesterone to help feed your own cortisol production. DHEA-S lab results are part of this picture. I do not know anything about hydro-cortisone dosing. You do not want to hard shut down your own production. Progesterone will be small dose. You can get 3% progesterone OTC at some vitamin shops, but not corporate shops. Look for KAL brand. Progesterone feeds cortisol production. I do not know what men should have for progesterone labs in an adrenal context. See page 220 in the book [below].
You can also try some pregnenolone caps [50-100mg] from a supplement store. This can help the brain directly. If on TRT without hCG your testes, the largest single source of pregnenolone, is shut down. Pregnenolone is needed to be made into DHEA in the adrenals, then to progesterone.... cortisol.
Lack of magnesium can lower progesterone levels. Get some magnesium supplements and you should be taking a supplement with trace metals as all enzymes contain metals as the catalyst agent.
If T3 is a bumpy ride, then it might be better to not take on an empty stomach.
This is a must read!
Note that chronic low level infections can contribute to or cause adrenal fatigue. Are your gums healthy? Any tick bites or Lyme disease? Unexplained joint pain or other aches?
Low vit-D status can be holding you down and limiting your ability to recover or respond to these hormone interventions. Try 5-6000iu ED in oil based caps. Some who do labs for Vit-D find that they need even more. Vitamin D3 is converted into [a steroid] hormone that every cell in the body has receptors for. All steroid hormones are transported to the nucleus of your cells where they effect DNA gene expression.
If you take statin drugs, CoQ10 supplements are vital, not just for your conditions.