MB 37: At the wrong time - For instance he mentioned "Ya'aleh VeYovo" when it was not Rosh Chodesh or Chol HaMo'ed, or he added [the Blessing] for Shabbos or Yom Tov on a weekday, and such like, and he realised after he had completed that Blessing or the whole of the Amidah.
MB 38: This is not an interruption - That would require him to go back to the beginning [of the Amidah] or to the start of the Blessing in which he had interrupted. But most Later Rabbis argue with this decision, and are of the opinion that this case is no better than where he interrupted by speaking in the middle of the Amidah - on which the Shulchan Aruch decides above in Siman 104:6 that he must repeat from the start of the Blessing or, in the case of the first three and last three Blessings, from the beginning of the Amidah; and see above there where we have explained all the details of the laws of speaking [in one's silent Amidah]. The Derech HaChaim and the Chayei Odom also concluded that the law is as these Later Rabbis said; and see in the Derech HaChaim where he adds more and writes that what we have said [in the name of the Later Rabbis] only applies where he mentions [in the Amidah] something that is inappropriate for that day, eg. he says "Zoch'reinu" etc. ["Remember us" - the first words of the additional verse said in the first Blessing during the 10 Days of Repentance] and "VeKos'veinu" etc. ["And write us" - part of the same verse] when it isn't the time for writing ["Kesivah"] that day [because it is not the 10 Days of Repentance], or he said "Ya'aleh VeYovo" through the words "On this day, Rosh Chodesh" [Yom Rosh HaChodesh HaZeh] which is patently untrue - because today is not Rosh Chodesh - and therefore it is as if he spoke in the middle of the Amidah as mentioned above. But if he did not say something untrue, eg. he said "Zoch'reinu LeChaim" ["Remember us for life"] but did not say "VeKos'veinu" ["And write us"] or Havdalah in the Blessing [that ends with] "Chonen Hado'as" [Gracious Giver of wisdom] where he mentions how Hashem, may He be blessed, has endowed him with a heart to understand and discern, and it is appropriate the whole year round to give thanks to Hashem, may He be blessed, for having given him a heart to understand and knowledge to acquire; or he said "Ya'aleh VeYovo" up to [but not including] "On this day of Rosh Chodesh" which is not then something untrue as he is merely adding to his prayer; and even though it is forbidden to add anything in the first and last three Blessings [of the Amidah, and "Ya'aleh VeYovo" comes in the last three,] nevertheless [adding a prayer as we have descibed] is not like talking. End of the quote from the Derech HaChaim. See in the Chayei Odom Chapter 25 where he writes that whether in the case of one who spoke in the Amidah unintentionally or whether in the case of one who mentioned [in the Amidah] something inappropriate to that day, after the fact if he did not go back to the start of that Blessing [where he interrupted], since he has completed the Blessing even though he has added something irrelevant to it, he is not now permitted to go back [to the start of the Blessing and say it again correctly], and all the more so [may he not go back] if he has already finished his Amidah; see there his reason in the Nefesh Odom.