Hi,
Welcome to HCM. I had gone through your question and understand your concerns.
Before starting medications we should know the cause of hair growth. 
The cause for post menopausal hair growth in chin are:
Previous poly cystic 
ovarian disease, adrenal disorders, ovarian tumours which produce male hormones.
The common investigations we do are serum 
testosterone, DHEAS, 17 
hydroxy progesterone, 
thyroid function tests, cortisol, ultrasound/CT scan of abdomen etc
Treatment:
If there is a specific cause from the investigations - the cause has to be treated
If there is no identifiable cause treatment will be
1. Anti androgens : finasteride, aldactone
2. Local : creams - Eflorinithine, 
laser treatment, electrolysis 
Hope I have answered your question. If you have any further questions I will be happy to help.
Regards
Dr Ajish TP
Consultant Endocrinologist