Should I inform my old supervisor of new insights that lead me to think his hypothesis must be adapted?
Definitely seems like something worth mentioning.  Regardless of whether this leads to any edits in the final version of the paper, presumably your old advisor cares about the question for its own sake and not just for publications and so would like to know if there's new information that pertains to your results.

Your old advisor will be in a good position to decide whether any changes need to be made, and if so whether it's fine to just wait until the reports come back for the minor changes.  Sometimes it's fine to have a paper that reflects your understanding at the time it was written.  Sometimes you can just add a small remark in the final version mentioning that new things have come to light.  Sometimes it might be that you should update the referee immediately.  It's hard to know without knowing more about the details of the situation, but usually it won't require pausing the review process.