When I restarted a character, I just deleted all the save files in the MyDocuments->MySaves->Skyrim directory. It means that number is in there somewhere; although I haven't cared enough to figure out if it's storing it in the savegame, or if it's counting the savegames in that directory.
Somewhere in the "new game" process it creates a 001 save file whether you manually save the game or not.