Sadly Apple haven't provided a way to completely transition an old iTunes library into the new format. Here is the standard layout:
~/Music/Music is the default location of the library, ~ being the user's home folder
~/Music/Music/Music Library.musiclibrary is the library package, replacing iTunes Library.itl et al
~/Music/Music/Media is the media folder, previously iTunes Media or iTunes Music
~/Music/Music/Media/Music is a hangover from iTunes, and contains all artist folders *
You can move and rename your old iTunes Media folder as ~/Music/Music/Media, then delete the iTunes folder. Make sure that Music > Preferences > Files has adjusted accordingly, or reset if needed. The Automatically Add to iTunes folder is redundant too.
tt2
* If you really want to eliminate this extra layer let me know and I will explain how.