You can add the github repository as a remote to your checked out gist repository.
$ git clone git://gist.github.com/1234.git
$ git remote add github git@github.com:ChrisJamesC/myNewProject.git
Push it to initialize the git on github
$ git push -u github master
This will also change the upstream of the branch, so github will be default.
You now can rename the remote of gist:
$ git remote rename origin gist
Each time you make changes (or pull changes from github/gist), you can do:
$ git push # To github
$ git push gist master # To gist
This will also push back your changes to the gist and not only the github repo.