While the Mirena Coil hormonal release is going to be in the normal range, it is possible that it throws off the internal balance sufficiently to cause the hair loss.
A possible situation would be a shortage of sex hormone binding globulin (SHBG). At any point 99% of hormones are bound to SHBG. The extra hormones released by the Mirena free up excess testosterone (yes, women have a little testosterone and men have a little estrogen) which generates blood vessel constriction in the scalp and results in hair loss just as it does for men. To make this worse, eat plenty of red meat, which raises testosterone levels by 20% within two hours after a meal. It also would be acclerated by drinking alcohol, which causes a fluctuation in estrogen levels.
The deciding factor is how sensitive your body is to hormonal fluctuation. Many patients do not experience symptoms on the Mirena because they are fine with great fluctuations.
I hope this is helpful. It is for informational purposes only, and does not in any way alleviate your need to have a local doctor that can advise you.