You could have 2 upstreams set in your local directory. Sort of similar to how you'd have a forked-repo set up. here's an example
$ git init
$ git remote add public https://username@github.com/username/public.git
$ git remote add private https://username@github.com/private/private.git
$ git pull private the-branch
$ git add -A
$ git commit -m "moving private to public"
$ git push --set-upstream public master