Cover off? What's wrong with that?
"The rain has stopped and they are taking the covers off."
OK, Aggers always uses it in the plural. I don't recall hearing anyone use it as a singular, if that's what you mean.
Or, "Mr. Brearley has moved Extra Cover off to the right and made a gesture that seems to indicate that he would like the sun to move a little squarer... Who is this man?"