Starting anew immediately reminds me of a practice of my second grade ("grade two" in Canadian) rebbe, Rabbi Hyman ז"ל. Thanks to him, I am allergic to finishing a parshah, a perek of gemara, a sefer, without starting a new one. Whenever we made one of those siyyumim (celebrations of completion) that second graders perform from time to time, he had us read the first line from whatever book we were starting next. It made enough of an impression that I still do it, 30 years later.
I used to annoy my Daf Yomi crew with the practice, too; we would finish a masechta, and they wouldn't have the new volume with them and the day's Daf was concluded, but I would insist on starting the first mishnah in the new volume.
Theoretically, there is no reason to do this for the end of some specific section; we could add another line at every pause from our learning, whether at the end of a book or mid-page. One could even argue that it's a negative practice, keeping us from spending time thinking about what we have just completed. But from a psychological standpoint, it's important to do this specifically when we complete something, so that we don’t see our study as complete, but as leading to something new.
The same could be said for starting new ventures in life whenever an old one is complete, having nothing to do with study. [This has particular value when the 'completion of the old' is associated with grief and loss, but that's a discussion for another time.]
The model for Rabbi Hyman's practice might be Simchas Torah, when we start the Torah with Bereishis immediately after we finish the Torah with v'Zos haBerachah, but just the other day I came across another, earlier basis for the practice:
The gemara (Avodah Zarah 19a) records a story involving two sages, Rabbi Shimon bar Rebbe and Levi, who were studying from the same scroll. They finished learning one book, and then they wished to start a new one. The story is only recorded in the Talmud because of their debate as to what they should learn next; the fact that they had just finished a text is not directly relevant and doesn't seem to belong, and so Rav Shmuel Eideles (Maharsha) comments:
מדנקט לה בכי האי גוונא דסליק ספרא נראה דאשמעינן שבשעה שמסיימין ספר אחד יש להתחיל ספר האחר
From the fact that it brought the story in this manner, mentioning that they had just finished a book, it appears that they were teaching us that when we conclude one book we should begin another book.So there's a solid source. May your every completion – secular year or individual day or study session – lead directly into a new beginning.
PS - The Maharsha adds a note:
כמו שאנו עושים בשמחת תורה ע"פ המדרש מפני קטרוג השטן
This is like our practice on Simchas Torah, based on a midrash of preventing the Satan's accusation.I don't know where this midrash cited by the Maharsha appears, but Rav Ovadia does quote it to explain why there is no kaddish between completion of the Torah and the start of Bereishis on Simchas Torah (Yabia Omer 4:Orach Chaim 22).
