In my siddur, it’s brought down that after Mussaf is said, Shachris isn’t repeated.
147: Rosh Chodesh Elul is on Shabbos and Sunday (2 days)
Sam2: One may put on tefillin again as long as his body is fairly clean and he has the proper mindset. There are those who argue not to put on Tefillin after Musaf (on Rosh Chodesh) until Mincha time. See also the Bais Yosef, Orach Chaim 44 where there is a practice to put Tallis & Tefillin on every fast day in the afternoon, to make up for missing Brachos (Moroccan practice).
Choppy: The period between Chatzos HaYom and Mincha Gedola (30-40 minutes) is usually considered Tefilla-free. I am mesupak if any Shmonai Esrai (save for Mussaf) can be said in that timeframe.