The prefixes "pre-" and "post-" refer to events before and after. For instance, "pre-season" and "post-season" or "pre-study" and "post-study". Is there a prefix fitting this pattern which means "during" that can fit in the form "(prefix)-season" and "(prefix)-study"? "Mid-season" and "mid-study" seems to refer to the middle point, so it does not seem like a good fit.
It varies from word to word, depending on the specific example, but the most common prefix is in-, or for there to be no prefix at all:
I can't wait for the season to start. The in-season games are so much better than the preseason ones!
After a bit of pre-study work, we did some studying, and then went for post-study drinks.
What about peri- as in perinatal. It does mean "around, about", and if used in a sense of time (as in perinatal) this could mean "during"
In medical terms, it's "preoperative," "intraoperative," and "postoperative," with "perioperative" encompassing the entire surgical process: before, during, and after (according to Wikipedia: http://en.wikipedia.org/wiki/Perioperative).
What I mostly see is the mid- prefix as in mid-season.