An unusual way to transfer your music, which works surprisingly well, is to use the import feature of Windows Media Player - or iTunes - or any other favorite application. Install the application, if not already there, and configure either a network shared drive or another external drive folder as the library location. Even a thumb drive would work well, and some MP3 players work as USB drives.
Then use the Add to Library function, or Import Media, etc. from the software player to search the source hard drive for all your songs. As it pulls the songs in from the source disk, the media player will build it's library on the backup destination hard drive... you even get the benefit of a well organized folder tree built from the song tags, if you're using any of the typical audio formats.
This works pretty well for when you don't really know all of the file locations on your computer, because the media player does the searching for you.