Push to a remote origin on a subfolder of git repository?

The indirect, and snotty answer, is that git isn't a deployment tool. So you should use phing, or capistrano, or ant, or puppet, or chef, or ansible, or grunt, or just write some bash script, etc.

The short answer is git-subtree split. I'll go with this.

The Explanation

Subtree, even more so that submodules, can be confusing if you don't really know what's going on. Normally they are used for modular development as an alternative to submdodules. I won't go into that because thats not how you would use it in this case.

For your purposes, you would perform whats called a subtree split to create a branch that contains only the directory you want. The subtree split command looks through all your repository's history for commits that effect files in a given directory and separates them from changes that don't. You're then able to put those commits (or a squashed version of them) to their own branch. As a consequence, when you checkout that branch, the given directory will be at the root, and nothing else.

Nuts and Bolts

I'm going to make some choices for you in my explanation, but you're free to change those. I'm designating the deploy branch, and I'm deciding to squash your subtree history into a single commit since you should never use that branch to do any development. I'm also assuming your repository name is origin.

git subtree split --prefix=wordpress --squash --branch deploygit push -f origin deploy:deploy

By pushing with the deploy:deploy notation, you dont actually have to checkout the deploy branch (which might take time to do) in order to push it. You could actually avoid even creating the branch all together by just pushing to the remote branch directly.

git subtree push --prefix=wordpress --squash deploy master

I recommend using this last method, since it avoids accidentally doing anything with the squashed subtree branch.

But,... subtree doesnt work for me because XYZ

Yeah, subtree splitting is still spotty sometimes. If for whatever reason the subtree command isn't an option for you, thats ok. You can use the older, even more confusing git-filter-branch command. See here for a full explanation on that command.

If your stuck with this choice, then.. that stinks. You're probably better off just using submodules (probably the way to go if your at this point), or writing some bash script to copy files to a different repo on you local and pushing... or something else terrible.

Or you could just use a real deployment tool.