If you want to transfer the local music from your Windows or Mac PC to your iPhone without iTunes, AirMore will be your preferable choice since it is a web-based application that does not require installation, signup or even any fees. AirMore app is fairly lightweight (3MB), which does not occupy your local disk.

Follow the steps below to transfer music to iPhone without iTunes

1. Download the app on your iPhone.
2. Navigate to AirMore's website. Aim your camera into the QR code scanning frame. Click "Agree" to create connection.
3. Simply tap on "Music" icon on the left side, click "Upload" to import your music videos from local database. Or you can also drag and drop files for uploading.
4. Wait until the process finishes.

Note: Your device and PC should be running in the same WiFi network.

The time I will use in downloading this; I will use that time to transfer the music with iTunes.

FonePaw iOS Transfer is a good choice. This program allows you copy photos, music, contacts, playlist, movies, TV shows and Podcasts etc. from PC to iPhone. Below are some guides, you can read them for more details:
How to Transfer Music from Computer to iPhone
How to Transfer Music from iPhone to iPhone

Tenorshare iPhone care pro is a good tool for you to transfer iPhone data to computer, it is also a good tool to clean up iphone space. Most iOS users like to use it to speed up their iOS device !
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For your information, FoneTrans for iOS can make this happen to transfer music to iPhone without iTunes. Besides, if your deleted your music or other data like deleted messages, photos, notes, videos, contacts, etc from your phone. You can use FoneLab for iOS, a professional iPhone data recovery software. You can have a try.

For transfer music from iPhone to iPhone, I am using the TuneFab WeTrans which consists of lots of functions, like transfer music, videos, photos and etc with the 2 expanding functions. Like it more than I can said.