You don't have to worry about suppressed TSH, taking any form of thyroid meds means that your pituitary gland detects that thyroxine is there so it doesn't need to send a signal to the thyroid to produce any, that signal being Thyroid Stimulating Hormone (TSH).
In 'Treatment Options' on ThyroidUK's main website, there is an article by Dr Toft, past president of the British Thyroid Association and leading endocrinologist, in which he states
"The appropriate dose of levothyroxine is that which restores euthyroidism and serum TSH to the lower part of the reference range - 0.2-0.5mU/l.
In this case, free thyroxine is likely to be in the upper part of its reference range or even slightly elevated – 18-22pmol/l. Most patients will feel well in that circumstance.
But some need a higher dose of levothyroxine to suppress serum TSH and then the serum-free T4 concentration will be elevated at around 24-28pmol/l.
This 'exogenous subclinical hyperthyroidism' is not dangerous as long as serum T3 is unequivocally normal – that is, serum total around T3 1.7nmol/l (reference range 1.0-2.2nmol/l)."
So it's FT3 that is the most important test.