file attachments reuse

I'm working with a site that has many, many pages that each have many, many files attached. I've now been asked to look at reorganizing some of the nodes and attachments. Straightforward enough. However, some of those files should/could be attached to more than one node. I really don't want to upload multiple copies of the same file.

Is there a way to attach the SAME file to more than one node? The File Attachments dialog has only an upload button. Is there a variation that has a search or browse server button?