First off, I used the wrong word. It’s apparently vowel raising and tensing, not narrowing. I got my words mixed up.
Anyways, the sound I’m talking about is somewhere between the a in cat and the e in bet for me. (IPA is unfortunately not that helpful, since it doesn’t acknowledge such a vowel.) I use it for any word that has -am or -an in it. I find this change extremely common in General American, to the point that a lot of people don’t notice it.
The exact vowel does vary, however. That “hillbilly” way I mention above of saying “may-uhn” for “man” is the extreme version of it. The further north you go in the US, the less narrow it seems to become.*
It’s something I noticed in school choir. We were taught this very open a (as in cat) vowel. And I noticed that it sounds pretty normal for most words, except those with -an or -am.
*Except in the Great Lakes region, where a in other positions starts raising and tensing. You can read more about it here.