You cannot git push something that is not committed. The solution is to commit often (and do it over ssh if you forget on a remote system). It doesn't need to a presentable commit. That can be cleaned up later. I use `git commit -amwip` all the time.
Sure, you might neglect to add a file to your commit, or commit at all, but that's a problem whether you're pushing to a central public git forge or not.