I once asked a rabbi about the purpose for the "Olat Ha-Tamid". He said it was for the nation’s sins of ignorance - the Shacharit for sins during the night, and the Mincha for sins during the day. Now I’m hunting for the rabbinic sources for that answer. Can you help?
The basic idea is in Midrash Raba bamidbar Parsha 21, as well as several other midrashim (see Pesikta d'Rav Kahana ch. 5) The Talmud Yerushalmi (Yumah ch. 8 p.45, Shavuot ch. 1 p.23) associates the olah with atonement of sinful thoughts.