Important update!
Please follow steps 1-5 as above, then skip this post and follow steps 6-9 here.

The steps in this post do not yield the desired results.
I am leaving this post to demonstrate the dodgy results it could lead to (here is another example).

Step 6
Check the address bar of the newly opened tab or window.
Look for "attachmentid=xxxxx" in the URL string, where "xxxxx" is the numerical attachment ID. At the time of writing, it was in the 40 thousands.
Copy the attachment ID.



Step 7
Use the "attach" tag to refer to the newly uploaded attachment.



Step 8
That's it!