How do I add music files to my iPhone without iTunes?
Sadly, but there is no way you can add music to Music app on iPhone without iTunes unless you have a jailbreak (as far as I know there is none for 10.3.3). Songs are added to some database, thats why you can't just put your song there and expect it appear in your Music.
What you can is to use 3rd party apps (music apps, I guess), that have access to their documents from a computer (since iOS 8.3 users are not allowed to access documents of any application). But then you will need to listen your music in that app you used.