Multisite 3.6.1 – Image Paths Contain /files instead of /uploads – How to Fix

(I posted this in another thread, but with a less appropriate Title, and couldn’t delete the thread)

I am running WP 3.6.1 (MU) on a new server.

Rather than migrate my network of sites, I exported my posts only from the old server using the Tools -> Export -> Posts – [XML file] method. I successfully imported the posts into the new sites on the new server. Since most of the images on the old server weren’t attached to the posts, I didn’t bother checking the box for downloading attachments.

The problem is that the imported image file paths are all like this:
/files/2013/09/Smokehouse-on-Shelby-and-Fountain-Square-Theater5.jpg

But should be like this:
/uploads/2013/09/Smokehouse-on-Shelby-and-Fountain-Square-Theater5.jpg

(All of my images on the old server were contained in wp-content/blogs.dir, which is not present on the new server.)

I have confirmed that when I upload a NEW image (test.jpg) it goes into wp-content/uploads/sites/60/2013/09/test.jpg.

I uploaded all of the image files from the old site which corresponded to this same year (2013) and month (09) into this same folder, yet these images don’t even appear in the image library.

Can anyone explain what is wrong?