The Gemoro on Chagigah 9a discusses the tashlumim of Chagigah on Shemini Atseres. In the first approach (Reb Yochanan), we learn a Gezero Shovo from the last day of Pesach. How does the Gemoro know (at this stage) that there is Tashlumim for the whole of Pesach?
[Later, the Gemoro learns alternatively from a Beraisa which does learn from a posuk that there is Tashlumim for the 7 days of Sukkos, before discussing Shemini Atseres. Does the Gemoro assume this posuk also applies to Pesach. If so, shouldn't it say so! Alternatively, is it a basic assumption that the Chagigah can be brought on any part of the festival?]
The Mishnah says that one who did not bring the Korban Chagigah on the first day of Yom Tov may bring it on any day of the festival, even the last day (Shemini Atzeres). The Gemara asks what is the source for this, and it seems that you are understanding the Gemara to be asking for the source for Tashlumin on all seven days of the festival.
RASHI (DH Mah l'Shevi'i), however, clearly implies that the Gemara is not asking for the source for Tashlumin on all seven days of the festival, because the source for that is obvious -- since the verse states explicitly that "it shall be for a festival for seven days," it obviously has Tashlumin during those days (as the Beraisa that you mentioned says). Rather, the Gemara is asking for the source that Tashlumin may be brought even on the last day of the festival -- Shemini Atzeres, for that day is an independent Yom Tov with regard to several matters. In response, Rebbi Yochanan derives from the word "Atzeres" in the verse that Tashlumin may be brought even on Shemini Atzeres, and the Tana of the Beraisa proves it from the verses of "ba'Chodesh ha'Shevi'i Ya'asu Oso...."
Mordecai