Why does the Torah write "Mil'vad Olas ha'Boker"?
Rashi: The Pasuk means that he brought the people's Minchah after the Tamid shel Boker.
Ramban, Moshav Zekenim (both citing Toras Kohanim) and Seforno: It means that he brought the people's Minchah in addition to the Minchah of the Tamid. 1
Ramban, Moshav Zekenim: It means that he brought the people's Olah in addition to Olas ha'Tamid.
Ramban (citing Toras Kohanim): The people's Minchah neither replaced the Minchas Tamid nor preceded it. .
Why did the Torah not see fit to teach us by the seven days of Milu'im, that the Kohanim's Olah did not replace or precede the Olas Tamid?
Ramban: Those were Korbanos Yachid to inaugurate the Kohanim. It is obvious that it would not replace the Tamid. 1
Moshav Zekenim (23): During the seven days of the Milu'im they offered only Korbanos Yachid, but not Olas Tamid or Ketores.
Unlike here, that the Olos, like the Korban Tamid, came in honor of the day. (EC)
Why did they offer the Minchah before the Shelamim? A Mishnah (Zevachim 89a) teaches that Damim (animals and birds) precede Menachos!
Panim Yafos: This was a Hora'as Sha'ah to offer the Mishnah first.
Meshech Chachmah: Also the Tzibur offers Menachos from which Kemitzah is taken (the Omer and Shtei ha'Lechem), but it does not offer Shelamim of Kodshim Kalim, therefore the Minchah came first. 1
In spite of this, the Mishnah says that Damim come first! Perhaps here is different, for both were Korbanos Tzibur, and the Minchah was like Kodshei Doros, so it was like Tadir. (PF)