Is one allowed to wear a hat cover because it is raining where there is no eruv?

The Poskim dispute over this issue Rav Moshe Feinstein (Igrot Moshe OC 1 108-110) holds since it is only a protection to the hat and is not part of the hat and as such can’t be carried. Rav SZ Aurbach (Shmirat Shabbat Kehilchata 18, footnote 46) holds that since people don’t remove it when it stops raining it is a clothing artifact as the hat is and as such can be carried out of the Eruv. One can rely on the lenient opinion.