I think Eliezer probably meant us to know that Quirrell was H&C the very first time he appeared.

I disagree with this at least as strongly as you believe it. I'm pretty sure he meant Hat and Cloak to be a giant question mark. Hence the elaborate descriptions of the broad hat, the dark mist, and the gender-concealing cloak, all drawing your attention to the mystery of his identity. There have been hints, but they don't all point in the same direction the way they do for Quirrell and Voldemort. Some are red herrings. I conclude that we're not meant to be certain of who he is. We're meant to wonder and doubt.