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Generally when any medication states that side effects are temporary, it means the side effects will be experienced as long as you are taking the medications and such side effects are reversible shortly after the medication is stopped or out of the body system.
One of the side effects of levothyroxin is hair loss. With the statement above, temporary hair loss would mean hair loss while you are taking the levothyroxin. However, be aware that symptoms of thyroid disease may include hair loss. That would mean that hair loss may be experienced continuesly if the thyroid condition remain untreated even without taking the levothyroxin. (This is only to explain. I believe you will need levothyroxin as long as you have the same thyroid problem exist. )
I am not sure if Minoxidil can do any help but I am afraid it won't make the condition any better. It is not contraindicated in thyroid conditions though but Minoxidil has to be used with precaution in the presence of hypertension or heart,kidney and liver diseases. You should discuss it with your doctor if you feel like trying it.