There is a story brought down that Rav Moshe Soloveitchik, the son of Rav Chaim Soloveitchik, was once up to Shemoneh Esrei in Shachris, when the tzibur was about to daven Musaf. He reasoned that he could daven Musaf then and daven Shachris thereafter. This way he would still be davening Musaf together with the tzibur. He also felt that he was also accomplishing being someich ge’ulah l’tefillah with the tefillah of Musaf.
According to what we have established there is no need to be someich ge’ulah l’tefillah during Musaf, and there is a requirement to do so by Shachris. Perhaps then it would have been advisable to continue with Shachris and have been someich ge’ulah l’tefillah to that tefillah, rather than switching to Musaf, which doesn’t require one to be someich ge’ulah l’tefillah. This is especially true since one can fulfill tefillah b’tzibur by davening Shachris when the tzibur is davening Musaf. (I do not recall the source for this.)