I would consider may and might nearly interchangeable if we are talking about how people possibly feel in the future. Another word that would work here is “some people could,” which is more likely than may but less certain than will or would (and even less emphatically certain than shall). Both “may well feel” and “might well feel” imply that they very likely do feel that way, and that they would be justified in feeling that way, but it’s still not certain.
However, if we are talking about the past, I think may and might are more distinct. To me, “may have felt” means that we are not certain whether or not someone felt that way in the past. The way I use it, especially in writing, “might have felt” is a counterfactual, and we are talking about a scenario that did not happen. The phrase “could have felt” can mean either. However, people often use these terms more loosely than that, and I’ll understand them.
When we’re talking about the present, it gets a bit more muddled. I use do instead of will when talking about the present. The word may still implies that it’s unknown whether they do or don’t. The words I would use for a hypothetical statement are might or could, but it doesn’t feel like an error to use those the same way as may (unlike with may have/might have).
So, we’ve covered statements about the future, like you asked about. If I say, in the present tense:
They didn’t think so the last time I talked to them, but they [might/may/could/may well/would/do] feel that way now.
There’s no if clause nearby to make any of these verbs contingent. Might implies that they probably don’t, could implies they probably do, and may is in between. If I say may well, I’m implying it’s almost certain, and would means I am certain of it, but still implies I have not confirmed it. If I say do, it’s a statement of fact. You cannot use will in this context.
Talking about the past:
They don’t feel that way, but in the past, they [may have/could have/may well have/would have].
In this context, many native speakers would also use “they might have,” but that sounds wrong to me, and I would only use might have , when writing or speaking carefully, in a sentence like, “... but under different circumstances, they might have.”