It says in the Mishne Brurua to wear a hat and jacket by davening and bentching. Hilchos Tefiloh, Siman 91, sk 4, mb 12 (megulah)
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 OC 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.)