CHERRYBIM, alleviate your amaratzus I’ll re-point to you the Mishna Berura that requires a JACKET during davening that has been mentioned previously on this thread:
S’O Orach Chaim Hilchos Tefiloh, siman 91, sk 12 (megulah) [sk 4, mb 12 ?]
Mishnah Berurah 8:4, citing the Ba”ch, requires two head coverings. (Shulchan Aruch O”Ch 282:2)
Please learn it and report back!