Strategies for managing design changes after site is live

I’m hoping someone can help me with some real-world advice about a way to handle overlapping design changes on a live site. I guess what I’m missing is the webflow equivalent of either Git branches or selective upload via FTP. I’ll give you an illustration that happened to me this week:

Client site is live - standard brochureware site. We have a package of work to restructure some of the pages which will take a couple of weeks to complete. So, I start making those changes to the site and push them to the staging server for review/iteration. In the meantime someone spots an issue on the live site that needs fixing. A couple of classes needed renaming so I quickly make that change. Now I can’t push the fix to the production server without the unfinished site restructure work going live as well. In Git I could have used branches or with old-skool FTP I could have just uploaded the relevant files (3).

What do you guys do to handle these situations? I’m assuming there’s a clever “webflow way” of handling this situation I haven’t spotted so if anyone can point me in the right direction that would be great.