the easy way is to remove the HDD from the first computer
and putting it in the second computer (atleast temporarily)
as a secondary HDD to copy files to wherever you really want them
What shines for a task like this is a file management program
like Z-tree, where you can do a "filespec search"
In the case of mp3 files "*.mp3" Tag all those files (Control T)
Then alt-copy (alt-C) them from the drive where they are to
the drive where you want them.
an "alt copy" in Z-tree will preserve or even create the same
folder/subfolder structure to the new location.
it will do this "unattended" once started, it will ask you if you
want to automatically overwrite duplicate files, and give you
severalways to deal with this occourance, Skip that file,
Overwrite only older version of files, etc...
It usually takes ~15-18min to copy ~35gb of mp3 files from
one HDD to another.
I do operations like this as a matter of routine, but I also have
one HDD in a server drawer, I cannot "hot swap" it, but I can
move a drive from computer to computer about as fast as I
can shutdown and reboot, because I have a mounting cage
in each computer.
I also have "Backup drives" that are normally NOT in my computer
I also prefer keeping my audio data on what some would call "pure data drives"
which are simply a HDD on which there are NO files other than data files