Get out of OS upgrade installer on restart?
I wrote a blog post on some options, and you should be able to boot to your Recovery HD and re-install some OS X onto an external drive. You could also try to re-instal the OS on top of the current drive as that's not designed to delete anything important and might just work to painlessly repair things.


http://apple.blogoverflow.com/2015/09/back-up-then-step-up-to-el-capitan/
http://apple.com/osx/recovery


Without knowing how proficient you are with more advanced install options, installing even on to a 16 GB USB flash drive would get you to a place to re-run the correct installer and try again with the upgrade. From the second link:


  If you use the Recovery System stored on your startup drive to reinstall OS X, it installs the most recent version of OS X previously installed on this computer