The reason my answer could have changed if you were already in treatment vs. not being diagnosed is that TSH can be influenced by thyroid meds, and that would invalidate your result of .56. Since you're not in treatment yet, my original answer stands. It sounds very much like "secondary hypothyroidism", which is caused by a faulty pituitary response.
Good endocrinologists are rarer than hens' teeth. I'm sorry the one you've seen wasn't more willing to see past the "normal" lab results. That's the way most endos seem to be.
I will tell you not to give up on this being low thyroid; but I also have to say you probably won't have an easy time getting diagnosed. That doesn't mean it's impossible, but you'll have to be persistent about it. Try seeing an osteopathic doctor (DO) instead of an MD, if you can. Or find an MD or DO who prescribes Armour Thyroid. They tend to treat more "outside the box" cases than doctors who only prescribe Synthroid.
Learn as much as you can absorb about hypothyroidism so that you can make a case for yourself to any docs you may see. Thyroid for Dummies
will teach you the basics. Solved: The Riddle of Illness
by Dr. Stephen Langer may give you ideas about how to get a diagnosis. When you see the next doc, make an argument about those low-normal free Ts. Say, "Yes doctor... I know that these are in 'normal' range, but maybe they aren't 'optimal' for me." That statement once made the difference for me, getting me a dose increase when my doc had said I should stay on the same one.
Five years ago, when I suspected I was hypothyroid, I asked my then-primary doc for a TSH test. He actually refused such a basic, small request. I insisted, and when the result came back high-normal, he said I wasn't hypothyroid and wouldn't treat it. The next doc I saw was an endo. To her credit, she made the diagnosis, but ultimately didn't offer a treatment that would work for me. I had to move on to my current doc to be adequately treated. It's a cruel world out there for thyroid patients, but it can be successfully survived. I once read an article by a medical doctor who said it takes an average of 6
consultations before a hypoT diagnosis is made. That should tell you how ignorant most MDs are about to this condition. It should also tell you that giving up too soon isn't in our best interests.
Hang tough and don't give up yet! I'll be keeping my fingers crossed that you'll find your answers soon.
Edited to add... That family history of autoimmune diseases is important. Even though you have tested negative for thyroid antibodies, it doesn't mean you don't have any. The tests have a 20% error rate... meaning that you may have tested false negative. I've seen several members here who were ultimately diagnosed with autoimmune hypoT, even after negative antibody testing. Don't let any doctor tell you that you can't have it simply by interpreting what can be a faulty antibody test result.