Also could make surrounding sound effects slightly louder while you are sneaking, so you can better hear people that may follow you.
The Music is going to be less Action oriented and mostly calm Ambience that should fit the Area.
Certain things like Songbirds in the Area could chirp to the BGM too, for example. Or Leaves bristling in a way that is in tune with the music but not too obviously, since this isn’t a Cartoon.