Feif Un:
Regarding the halacha requiring a Yid to wear a jacket and hat for davening and bentching, I believe it is derived from Chullin 138, from a discussion of the Kohein Gadol’s turban (the mitznefes). Also see the Shulchan Aruch O”Ch 282:2. Mishnah Berurah 8:4, citing the Ba”ch, requires two head covering for all of davening.
(Rav Herschel Schachter cites the Pishchei Teshuvah who in turn cites the Shlah that it is necessary from the point of view of atifah, being cloaked while being aware of G-d’s presence.)