on his head (i.e., he is not wearing a yarmulka), then we are not concerned about the possibility that he may carry his hat in his hand (M.B. ibid.).
The Magen Avrohom (s.k. 52), quoting the Maasas Binyomin, expands this halacha and writes that one who is always careful to wear a hat outdoors may wear a hat on Shabbos, for even if the hat were to blow off, he would make sure to replace it before walking further. Accordingly, one who is careful in this respect may indeed wear a hat outdoors, even over a yarmulka.
that he may walk outside while holding his hat, similar to jewelry
[Although there are numerous ways to define a reshus harabim, a lenient approach is used regarding this prohibition. Accordingly, most cities do not contain a reshus harabim, and a hat may be worn without concern.]
However, the Mishnah Brurah (301:154) brings the opinion of the Taz that this prohibition applies even in a karmelis. Furthermore, the Biur Halacha (303:18 d.h. ki) writes that ideally one should not rely on this lenient opinion. In a location with an eruv, a hat may be worn without concern.