[Steven Zahm]

I think I may have confused you, in a multisite setup, your old folder should have been:
/public_html/wp-content/blogs.dir/4/connection_images/

My suggestions is, yes, copy them to that folder so Connections can find them to move them to the new folder which would be:
/wp-content/blogs.dir/4/files/connections-images/{entry-slug}/{filename}

I’m guessing you’ve change the defaults uploads folder to be files. That’s fine. I just want to confirm so I know as much of your configuration as possible.