Drupal - How to move files already uploaded to a new subfolder?

It sounds like you're looking for File Maintenance.

Ever wanted to move files from your sites/default/files directory (or whatever you have configured) to a (new) subdirectory? Never did it because you knew it's a damn mess since you need do move the physical file, adjust the database files table and look through all the node_revisions table and other content to adjust the link to the file so as not to break your site?

That should cover any files that are managed under the files table (pretty much any field you should have). If you have any other files under sites/default/files that aren't managed by Drupal, then custom redirect htaccess rules should cover those files.