pushing to a remote branch that one does not own

Hello, I want to contribute via "push" to a branch which I have previously cloned from a remote repository. There is a problem:

If i create a new commit on branch "master", which coincides with
origin/master, then I can push this commit simply with
    git push
However, if i switch to a different branch of the repository which was
cloned, for example by doing:
    git checkout origin/lilt 
Then I get the message: 
"You are in 'detached HEAD' state. You can look around, make experimental
... (etc)"
So I HAVE to create my own new branch based on the one downloaded in order
to start committing and pushing changes. My question is therefore: 

Am I strictly limited to committing only on the master / origin/master branched, and forced to make a new branch for every branch that I cloned from the remote repo, or is there a way of making the heads of the remote branches visible as local too?

Thanks, 

Iannis Zannos