[RESOLVED] Apple Time Machine wackiness - am I missing something?

I've recently upgraded my Mac mini to a newer model and I need a smidge of help. Just before I got the newer mini, I backed it up along with an external drive. Once I set up the new mini, I transferred everything from the external drive to a new, larger one and erased the first drive. I've realized that there were a few files on the first drive I'd saved on a previous back up to my Time Capsule drive and would like to recover them (no big deal if I can't. Just some JPG photos I might be able to find on other sources online but nothing truly earth shattering). I'm having some trouble connecting to the first external drive in Time Machine because I think it's because I haven't run a formal first back up on the new mini (I think) so I'm wondering will I have to set up the old mini again and pull everything back in that way?

Thanks, though!

Hey, thanks for the suggestions, both here on the board and behind the scenes. I've gotten it worked out!