Same Time Machine HD with new computer

I just bought a new MBA and I want to use the same external HD for Time Machine that I used for my old computer. Time Machine works perfectly on my old computer, but when I recently plugged it into my new computer, it started backing up and now when I enter Time Machine on the new computer, I can only see today's backup. The thing is, I can still get to my old files by going Time Machine/backups/old computer/date of backup but it is a pain to do it this way and somewhat unintuitive. How can I restore Time Machine to the way it usually works on my new computer with all my old files?

p.s. Sorry if this gets asked a lot, but I Googled around and all I could find was stuff about Time Machine migrating data between Macs.

Edit: Did some searching on the Apple Communities and found this thread:

https://discussions.apple.com/thread/1953178

Pondini's post is particularly useful. If anyone could add anything that would be great!