How to clear new Music app's cache of streaming music on iOS?
Easiest solution I have found so far is to disable the iCloud Music Library and open the Music app so that it can delete the cached files and/or release that storage to iOS to clear when needed.

Details: This solution is an adaptation of solutions I found for the same issue with iTunes Match. The clearing and rebuilding of your music library takes time, so I hope Apple adds a simple "clear Music cache" button to the settings in the future.


Settings>Music: Turn off iCloud Music Library
Open Music app: Your library should start to clear
Settings>Music: Turn on iCloud Music Library
Open Music app: Your library will be rebuilt


The process preserved my offline copies of songs previously synced to my iPhone prior to switching to the iCloud Music Library. I am not sure if it preserves offline Apple Music copies.