Test for Hashimoto's thyroiditis which is the most common cause of hypothyroidism in developed countries. TSH, free T4, free T3 and thyroid antibodies - thyroid peroxidase antibodies - TPOAb and thyroglobulin antibodies TgAb.
I have also read this info: "Some women demonstrate a mixed pattern of speckled/homogeneous. These same antibodies appear positive in women with lupus, rheumatoid arthritis, Crohn’s disease and other autoimmune diseases." You might want to see a rheumatologist for a full work up.