Trying to restore a iPhone 6 backup to an iPhone 5s

I am trying to restore an iPhone 6 backup to an old iPhone 5s (I just left a job where I used an iPhone 6, and I need to use an older phone I have lying around until I can get a new one with my next job). When I try to restore to the 5s, the iPhone 6 backup does not show up as an option. I'm guessing that this may be because the iPhone 6 backup is too large to fit.

Is there a way to at least pull my contacts from the iPhone 6 backup so that I can sync those with my old phone?