No, it doesn't really work, I'm afraid. "At" might be correct, but there are other problems with your sentence. Can you explain a little more about what you're trying to say and in what situation you would say it? (For example, do you mean your hair is falling out?)
Even allowing for hyperbole, hair falling out at a "huge rate" is something I couldn't say (and I've often needed to say something similar because it's a particular problem of my daughter's). "At an alarming rate" is what springs to mind.
I agree that speed is implied here. Given that "a huge rate" sounds like an exaggeration because, as you rightly say, the amount of hair is limited to start with, I thought that "alarming" tends also to hyperbole and so retains something of the enormity of "huge" . "Alarming" is much more emotive, I agree, but people do tend to be sensitive about going bald.