I believe the correct size bit to drill out a .370” hosel is 9.4mm. In absence of having the correct size mm bit is 3/8” acceptable? Same question would apply to a .335 hosel. What would be close enough to a 8.6mm?
3/8″ is the drill bit sized used for a .370″ hosel bore. If you are drilling to change from .355″ taper tip hosel bore to a .370″ parallel I would recommend step drilling to keep from changing the hosel bore angle. I would first drill with a “T” size bit (.358″) for the full depth of the bore. The drill again with a “U” size bit (.368″) for the full depth of the bore and finally with the 3/8″ bit for the full depth of the bore. Using this method the bits will stayed centered in the bore and you should not change the bore angle.