Change name of file using regex after script creation

I’ve created a script which let’s a client upload an mp3 audio file however it’s being save in a format like 2030131_9ec3-4ad1-84ae-c98b68390322.mp3. The files reside in a folder and sub-folder format that resembles a date. ./2019/05/31/639486572_016b0f3d-9ec3-4ad1-84ae-c98b68390322.mp3.

The goal is to remove everything before and including the _. Which I’m able to do with rename -v 's/$\.+_//'

But when I try to couple this with find it seems to rename all the files found in the subfolder correctly but places\moves them to the root of the folder I’m working out of so in this case ./2019

The command im running is find -type f -name "*_*" | rename -v 's/$\.+_//'

How can I ensure the files location doesn’t change ?