From what I understand, a three-day zav must offer korbanos related to his ziva and cannot, until that time, eat any korbanos. A mishna in chapter 8 of P'sachim says that such a person whose ziva-korbanos have not yet been offered [but, commentaries add, has passed those korbanos to the authorities to be brought] can have a korban pesach slaughtered for him, as it can be for anyone who can eat from it. Why is this? The pesach was brought after the afternoon tamid which was otherwise the last korban of the day; this zav won't, then, have his ziva-korbanos brought that day and won't be able to eat from the pesach, so why is it being slaughtered for him? Or, to put it another way, Tif'eres Yisrael says they will bring his ziva-korbanos for him the same day, but how can they?
Tosafos in חגיגה daf 2b (כופין את רבו) already discusses this, and brings it as an example of doing a minor infraction to allow a fellow Jew to keep a great Mitzvah.
תדע דשרינן ריש תמיד נשחט (פסחים דף נט. ושם) להקריב לכהן אחר תמיד של בין הערבים כפרת חבירו כדי שיאכל קרבן פסח לערב אע''ג דאיכא עשה דהשלמה
If you look at the source - in פסחים daf 59a - the Gemara discusses this very question. Various answer are suggested.
One simple answer mentioned there is that birds are allowed to be sacrificed after the afternoon tamid - always. They are not considered to be "real sacrifices" - and this Mishna [only] refers to a Zav who is eligible to bring birds.
אמר רבינא אמר רב חסדא הכא בחטאת העוף עסקינן שאין למזבח אלא דמה
Others learn from a verse that after the afternoon tamid one can bring the Pessach and the Zav's sacrifices.
ואין לך דבר שמתעכב אחר תמיד של בין הערבים אלא קטרת ונרות ופסח ומחוסר כפורים בערב הפסח שטובל שנית ואוכל את פסחו לערב