Should I inform my old supervisor of new insights that lead me to think his hypothesis must be adapted? I wrote my undergraduate thesis on the topic my supervisor was then doing research into. He now has a paper on that topic under review, which I had read and commented on while working on my thesis.

I am now in a graduate programme on another university, and for some course I needed a small project to write a paper on. After consulting with my old supervisor, I took a research question that is relevant in the light of his work, but also readable separate from it, because the teacher supervising this new project has -of course- not read the paper under review.

While working on the project I found new data that leads me to question a minor section of the article under review. I (think I) can't expect my old supervisor to go into much discussion on the topic - after all, he is not supervising my current project.

I want to stay on good terms with him, first of all because I hold him in high esteem but also because his research group is a serious option for a PhD position later. But should I inform him of the things I found out? On the one hand, I suppose it would be appreciated - on the other hand, the article is already under review. Also, the work is never finished, so finding new insights is part of life and we have to accept that publications are never perfect.

I have seen Should I email old supervisor about my progress?, but this is a more delicate situation.
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.