I had the same thing happen to me. When I was diagnosed with Hashimoto's my TSH was 1.57, but my antibodies were over 1,000!!!! I am now treated with 3 grains of Armour thyroid and feel better than ever. I would definitely request a full thyroid panel with Free T3 and FreeT4..the free's because those are the unbound thyroid hormones that your body has available for use. Make sure you also ask for antibodies to be checked!
Have you had thyroid antibodies tested? Hashimoto's disease has a genetic component, and almost always can be diagnosed by elevated TPO antibodies. You might want to ask your doctor to check. And I've read that if you have the antibodies, TSH isn't dependable. Also, the range for TSH now used by the AACE has a high of 3.3, but you're pretty close to that.
With the family history, would check thyroid antibodies -- if they are positive and the TSH is still >2.5 then given your symptoms a trial of thyroid hormone to get the tSH around 0.5-1.0 would be reasonable. There are no studies to prove this, but it may be worth a trial and I have had some patients benefit from this approach --- I would usually only start treatment in such a case if the thyroid antibodies (TPO and /or Tg) are positive.