(a) RASHI in the Mishnah (DH u'Par) and in the Gemara (70b, DH v'Achar
Kach) explains that the afternoon Tamid is brought after the *fifth*
Tevilah, following the Avodah of the removal of the Kaf u'Machtah (which
was done after the fourth Tevilah). The Tamid is offered at the same time
(that is, right before) the Ketores and the Neros. A clear source for this
opinion is found in the Beraisa (Tana d'Bei Shmuel) near the end of this
Amud that says that the Kohen Gadol brought his Ayil and the nation's Ayil
(Ayilo v'Ayil ha'Am) and the Emorei Chatas after the third Tevilah, "but
the Par ha'Olah, Shiv'as Kevasim, and Se'ir ha'Na'aseh ba'Chutz (i.e. all
of the remaining Musaf offerings) [were brought] *together with the
afternoon Tamid*." Since the Beraisa contrasts the times when the Ayilo
v'Ayil ha'Am were brought with when the afternoon Tamid was brought, this
clearly implies that the afternoon Tamid was not done at the same time as
the Ayilo v'Ayil ha'Am, but was done after the fifth Tevilah. (See CHART
#15, footnote 9, and Chart #16 footnote 1.)
(b) The RAMBAM (Hilchos Avodas Yom ha'Kipurim 4:2), though, rules that the
Korban Tamid is brought after the *third* Tevilah, together with (i.e.
right after) the Musaf offerings. The RAMBAN (Vayikra 16:32) adduces proof
to this opinion from the wording of our Mishnah, which -- when discussing
what was done after the fifth Tevilah -- mentions only the Ketores and the
lighting of the Neros. If the Tamid is also brought after the fifth
Tevilah, then the Mishnah should have mentioned it, since it mentions the
other routine, daily Avodos done at that point. On the other hand, if the
Tamid is brought after the third Tevilah, the Mishnah does not need to
mention it because it anyway leaves out a number of Avodos done after the
third Tevilah.
Further support for this opinion may be adduced from the next Beraisa cited
by the Gemara, which says that the Se'ir ha'Na'aseh ba'Chutz is brought
before the Ayilo v'Ayil ha'Am, after the *third* Tevilah. The Beraisa
continues and says that after the Ayilo v'Ayil ha'Am, the Emorei Chatas are
burned, and afterwards, the afternoon Tamid is brought. This implies that
all of these Korbanos are brought after the third Tevilah. (Rashi, though,
explains that when the Beraisa says, "afterwards, the afternoon Tamid is
brought," it does not mean at the same time as the previously mentioned
Korbanos, but it means much later, after the *fifth* Tevilah.)
Furthermore, the Mishnah says that the Se'ir ha'Na'aseh ba'Chutz is
brought with the afternoon Tamid. According to Rashi, it must be that the
Beraisa and Mishnah are arguing; the Mishnah maintains that the Se'ir was
brought after the fifth Tevilah, and the Beraisa maintains that it was
brought after the third Tevilah. (See Tosfos DH v'Achar Kach, and Chart 16
footnote 2.)
How does the Rambam understand the first Beraisa (Tana d'Bei Shmuel), which
we cited above as a proof for Rashi's opinion? That Beraisa clearly says
that the Korban Tamid is *not* brought during the same Tevilah as the Ayilo
v'Ayil ha'Am!
RABEINU CHANANEL (cited in TOSFOS DH d'Tana) has a different Girsa in that
Beraisa. The version in *our* Gemara says, "He then brought the Ayilo
v'Ayil ha'Am and the Emorei Chatas [after the third Tevilah], *but* the Par
ha'Olah, Shiv'as Kevasim, and Se'ir ha'Na'aseh ba'Chutz [were brought]
together with the afternoon Tamid." The version of Rabeinu Chananel reads,
"He then brought the Ayilo v'Ayil ha'Am and the Emorei Chatas *and* the Par
ha'Olah, Shiv'as Kevasim, and Se'ir ha'Na'aseh ba'Chutz together with the
afternoon Tamid." That is, all of those Korbanos, including the afternoon
Tamid, were brought after the third Tevilah. This is likely the Girsa which
the Rambam had as well.