Q. I also wondered, why is this time period chosen for Pirkei Avot and not another time?
A. Because of its contents, the name is sometimes given as Ethics of the Fathers. Pirkei Avot consists of the Mishnaic minor tractate of Avot, the second-to-last tractate in the order of Nezikin in the Mishnah, plus one additional chapter. Avot is unique in that it is the only tractate of the Mishnah dealing solely with ethical and moral principles; there is relatively little Halacha (Jewish law) in Pirkei Avot.
In most communities it is said at least until Shavuot, however many continue beyond Shavuot right through to Rosh Hashanah.
While the core season of this study is in the spring, many traditional communities (such as the Chabad movement) extend this study cycle throughout the entire summer, reading the chapters again every Shabbat until Rosh Hashanah.
Rabbi A. Bartfeld as revised by Horav Y. Hirshman, Horav Dovid Pam, Horav Aharon Miller and Horav Chanoch Ehrentreu, Horav Kalman Ochs, and Horav Dovid Bartfeld consulting in need Horav Hagaon Rav Yitzchak Berkowitz Shlit’a