Rain Hat cover on Shabbos

Igros Moshe (OC I #108) Question: Can a thin plastic hat cover be worn when walking outside on Shabbos to protect from rain? Answer. Any thing worn as a garment is permitted according to Nedarim (55) even if the intent is just to protect against dirt. But simply to cover the hat and not the body at all is prohibited. It is clear from the Taz and Magen Avraham that the meaning of “wearing as garment” requires at least partial direct covering of part of the body or head. The old practice of wearing a hat on a hat was to provide greater warmth and is thus permitted while a rain hat cover does not increase the warmth nor is it even for appearance and in fact looks worse wearing only the hat alone. Its only purpose is to protect the hat from damage which is prohibited. This is not the same as wearing galoshes which are permitted because they add warmth and thus they not only protect the shoes but also the person from discomfort of water soaked shoes and cold. In conclusion the wearing of the plastic rain hat cover is not permitted. This is what I tell all those who ask me.

It is important to know that others such as Rav Menashe Klein disagreed. Rav Ahron Kuler also disagreed but he said to all that asked that the halacha was that it was prohibited because everyone should submit to  the ruling of Rav Moshe