backup data from unbootable hard drive

The internal drive on my iMac would not boot. I used the command-R on restart then reloaded high sierra. When I tried to boot after that, it only went as far as the login screen and it would not take my password. I was able to boot the computer from an external drive and the internal drive showed up, with the directories and files appearing to be intact, but I imagine somehow the OS files had become corrupted.

My plan is to re-format the internal hard drive and install a new high sierra. Unfortunately my last time machine back up was several months ago. Since the files appear to be intact on the internal HD, is there some way to backup the files on that drive using time machine from the external drive so that I could restore them to internal HD? Could I achieve the same results with carbon copy Cloner?

Thanks for your help!