Move away content from public_html in a git repo on live server

I get the security thing about moving content away from public_html/. 
My workflow is that I push from my local machine to GitHub and to my live server. 
Yes, I can change the structure on my live server (like moving certain content up one level), but I guess I must keep it on my local machine. 
How do I push to my live server after having moved content up one level?