I Have Hypothyroid - Hashimotos I Am On Levothyroxine 112

I have Hypothyroid - Hashimotos I am on Levothyroxine 112 MCG. I get my blood work done every few months to monitor my levels when having issues.

About 15 months ago my hair started breaking around the face and receding on the sides in front. Than I started loosing a bit more hair than the normal 150 hairs a day for about 8 months than the last 3 1/2 months my hair has fallen out all over evenly about 50% of my hair and the texture got frizzy. The last 3 days I have lost a bunch of eye brow hairs.

I am gluten free and eat a very healthy diet, my blood work is in normal range. I am in Menopause at 49 years old but this is way to much hair to be loosing. I work out 4x week. I don't smoke or drink.

My doctor is at a lost but I feel it's my thyroid because I lost so much hair 5 years ago when my numbers were just a little off and no one wanted to give me medication. When they did my hair stopped falling out.

TSH 0.96 (normal Range)
T4 1.7 (normal higher end of the range)
T3 2.4 (low end of range)

Nov 2014 My hair was full and healthy my numbers were
TSH 1.25
T4 1.7
T3 3.1

If my doctor gave me T3 what would happen to my TSH and T4?
Can you suggest how to get my current those numbers up to the Nov blood work? I am very sensitive to my numbers being off but my doctor doesn't think it's my thyroid even though I have had all extensive blood work done and everything is normal. The dermatologist put me on Romaine for men. I feel like no one is helping me and so worried about the hair follicle. I go to Regenix in Beverly hills and I have been getting treatments for 5 years which helped my hair grow back healthy and full. When looking under a microscope the new hairs are thin and very weak and unhealthy.

Thank you very much!