git clone is a little bit overkill for this purpose.
The mechanism that git provides to deal with other servers is fittingly called remote.
This is typically automatically set up when you clone, explaining why that was your instinct.
Documentation can be found here:
http://git-scm.com/book/en/Git-Basics-Working-with-Remotes
Some summary commands:
git remote add [remote-name] [branch-name]
git fetch [remote-name]
git push [remote-name] [branch-name]
In addition, you might consider setting up tracking branches, which will eliminate the need to qualify the remote name each time you type a command. Documentation for that use case is here.
http://git-scm.com/book/en/Git-Branching-Remote-Branches
Usually, if you clone a repository, git executes
git checkout -b master [remotename]/master
Your tracking branch doesn't have to be master though.
Hope this helps.