To answer your question, cortisol assists in shuttling t3 into the cells. Without cortisol, thyroid will not function properly. Thyroid is responsible for the “heat” we feel.
From my experience, most of us in the “low temp” category have been under a chronic stress for sometime. The stress may have come from extreme dieting, overtraining, work and/or family stress or a combination of a few. I have taken cortisol personally without success. IMO, if you do not fix the underlying issue, recovery will not happen. Matt breaks it down pretty simple. Stop exercise for a month at least (physical stress), eat a surplus of calories (minimize starvation stress), and relax. By sleeping adequate amounts the adrenals are able to repair in a calorie surplus environment. Once the they are recovered, a sufficient amount of cortisol will be produced which will assist t4/t3 in raising the body temperature. Without sufficient cortisol or thyroid, optimal temp will not be reached. IMO, shutting down the adrenals through RRARFing and sleep, while minimizing all stress is the best approach. Pretty simple.