I can't give you a "professional answer", as I am trying myself to improve my English level; but I have studied the may/might difference just this afternoon and I can share with you what my grammar says about the topic.
Usually may and might are interchangeable.
However, we do use "might":
1) always, when the situation is unreal
(ex.: if they paid me better, I might work better (here "may" is not possible));
2) preferably, in daily speech, when we speak about possibilities in the future
(ex.: I might paint the kitchen purple);
3) in a formal or literary use, to speak about what was typically the case in the past
(ex: Years ago children might be sent down mines at the age of six);
In short, it seems that English-speakers will use "might" to speak about something that is unreal or more or less far in time. From that I would generalize and say that, for them, "might" must sound ...something further ... less sure or, better, more unsure than "may" ... but it is just my opinion.
I hope you find my comment useful. I also hope that an English-speaker give his/her opinion about what I have written and, in case, corrects any mistakes.
By the way, the grammar from which I have taken rules and examples is the "Advanced Grammar in Use" by Cambridge Press.
Bye for now!