How to update iPhone 5 to a different iCloud backup?

So I dropped my phone in my toilet (sigh… I know… DON'T put your phone in your back pocket…) And I bought a new phone. It wouldn't let me back it up to my iCloud that was updated yesterday because the phone didn't have ios update 8 or something (I'm updating it right now) So because it didn't have that I was unable to update to when icloud was previously saved.
Instead I had to update it to an icloud that was saved a few months back. Now I lost a bunch of photos and contacts and such. I was wondering if now that I updated it to ios 9 I could update my phone so that it's like my old one.

If that makes any sense… Please help!

Just sign out and sign in to the different account

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