I'll assume you have Hashimoto's thyroiditis since it's the most common cause of hypothyroidism. And so the condition that is suspect is Lupus since it's a listed cause of elevated GGT, fatigue and joint pain. The following info is from the book "The Everything Guide To Thyroid Disease" by Theodore C. Friedman, MD, PhD and Winnie Yu Scherer:
"Although the majority of people with Hashimoto's will not develop any other disorders, it's important to know what some of these autoimmune conditions are in case you do start to experience symptoms. Keep in mind, too, that you may be more likely to develop Hashimoto's if you have one of these other conditions.
Type 1 diabetes
Systematic Lupus Erthematosus (SLE)
Inflammatory Bowel Disease
Multiple Sclerosis (MS)"
I have Hashimoto's thyroiditis, autoimmune pernicious anaemia (was my first autoimmune disease), vitiligo, and alopecia areta.
Excerpt from Medscape - Gamma-Glutamyltransferase...
GGT levels are increased in patients with liver diseases in general, including the following:
Hepatitis (acute and chronic)
Liver metastasis and carcinoma
Alcoholic liver disease
Primary biliary cirrhosis and sclerosing cholangitis
Extrahepatic causes for GGT level elevation include the following:
Carcinoma of prostate
Carcinoma of breast and lung
Systemic lupus erythematosus
Congestive heart failure and chronic coronary artery disease: The level of elevation correlates with the risk of death secondary to cardiovascular disease.
GGT levels may be increased due to medications, such as carbamazepine, cimetidine, furosemide, heparin, isotretinoin, methotrexate, oral contraceptives, phenobarbital, phenytoin, and valproic acid.
Smoking may cause elevated GGT levels.
GGT levels are higher among blacks.
GGT levels may be increased in mononucleosislike syndrome (MLS).
Decreased GGT levels are associated with hypothyroidism and early pregnancy."