Well, I respond well to touch when I'm upset. It doesn't have to be a hug. Can be something as simple as a hand on my shoulder. And you are right, there isn't anything you can do to stop the crying...it's a storm and I, at least, have to weather it out until the end.
A soft voice helps, but really, not saying anything for a while is best. I know your instinct is to logically work through the problem, but I canNOT think rationally until I'm through with the worst of my emotional upset. After the worst has passed, then I'm usually willing to joke about what happened, figure out what went wrong and how to fix it