You might still be able to recover most or all of the files from either the old PC or the external hard drive by using a file recovery program like TestDisk, PhotoRec, or Recuva.
Try not to use the hard drive that you're wanting to recover files from; You'll want to minimize the possibility of overwriting the deleted files.
It might be best to run the file recovery program from a Live CD/DVD/USB flash drive. Hirens Boot CD 15.2 contains all three file recovery programs that I mentioned. Also, Parted Magic ver 2013.02.28 (the last free version) and Ultimate Boot CD 5.3.5 contain TestDisk and PhotoRec. I can get you links if you need help in that department.
...and in one fell swoop, no less. Used an external hard drive to transfer my files to a new PC, but I couldn't carry any huge files over because they needed to be archived, so I cut all the chess files and pasted them directly to the disk, then my USB cable got accidentally unplugged while some other files were being transferred, and as a result, I lost all of my chess training and learning material. ALL OF IT. 40+GB that had been collected over the span of some seven years... all down the drain.
I continued on playing online without any additional resources like that, but my game has been slowly deteriorating, and I need to get back on track with my chess improvement. The thing is, the sites that I'd been downloading from are all dead, and coaching is expensive, so I'm in a bit of a pickle there. Any good suggestions? Thank you in advance.