I'm not sure how b est to advise you, but I'll tell you what I know.
Hairs around my upper lip tend not to be soft and so I have to hoik them out with tweezers.
Every so often I catch sight of long-ish, soft hairs coming from under my chin (are these the sort of thing you mean?). I've always pulled them out with tweezers too and so far (over 10 years) they have not grown in any darker or stronger.
I suppose you could use a very gentle depilatory cream, but I can't say anything about that. Good luck - they will happen whatever. As I get older mine have become white, which makes them much harder to spot. :)