Yes, I'll explain how it works... In a perfect world, our thyroid produces, both T4 and T3, but mostly T4. Of the T4 our thyroid produces, most of is bound by protein and can't be used, which is why we test the Free T4, which is the unbound portion in our blood.
Free T4 isn't used directly, by our cells; it must be converted to T3, which is used by almost every cell in the body, but like T4, most of the T3 produced is bound by protein. Again, we test the Free T3 (unbound portion) to find out what's actually available for use by individual cells.
Of course, when we have hypothyroidism, for whatever reason, we don't have a perfect world and not everyone's body can perform the conversion process they way it's supposed to. The pituitary produces TSH when it senses that there aren't enough thyroid hormones to satisfy the body's needs. There needs to be a balance between Free T4 and Free T3, with Free T3 being higher in its range than Free T4 in its, in order to alleviate hypo symptoms.
Since Free T3 is the hormone that's used by individual cells, increasing that hormone would bring down the TSH, as well as, possibly, stop your hair loss and other symptoms. Sometimes, when we first start taking thyroid hormones, it takes a while for the body to begin the conversion process and for Free T3 to track Free T4 levels upward.
Sometimes, there's a malfunction in the conversion process and Free T3 never does track Free T4 up. In these cases, we have to add a source of T3 medication in order to provide adequate T3 for our body. This can be done by adding a T3 medication such as Cytomel or its generic counterpart, Liothyronine or by switching to desiccated medication such as Armour, NatureThroid, ERFA, etc. Desiccated medications are also called "natural" hormones, as they contain, both T4 and T3. Desiccated hormones are derived from the thyroid glands of pigs, which some think is more natural than taking medications such as Levothyroxine, Synthroid, Levoxyl, etc. T3 medications, including desiccated hormones, are not available in all countries, so depending on your location, that might not be an option.
Iron is necessary for the proper conversion of T4 to T3 and selenium has been shown to help with the conversion process also.
Your Free T4 level is higher than we'd, typically, recommend, at 70% of its range; most of us feel best with Free T4 at about mid range (50%). However, your Free T3 is pretty low, at only 36% of its range and it's recommended that Free T3 be maintained in the upper half to upper third of its range. Since you've only been on the medication for 1.6 months ( I put that at approximately 7 weeks), it could take some more time for your body to adjust and for Free T3 to track the Free T4 up or it could be that you aren't converting the Free T4 to the usable Free T3.
Your TSH is also too high, even though it's within the lab's range.
Once we've been hypo, the body takes care of the most important things first, such as heart rate, body temperature, etc. Unfortunately, hair loss isn't a priority so that's often one of the last things to be rectified when we start on meds.
You can try to talk your doctor into giving you a T3 medication to increase your Free T3 level, but it still may take some time for your hair to start growing back. It also takes vitamins/minerals time to work, so I know you want your hair back and it's hard, but you'll have to be a bit patient; it will take time.