Well, I also think it depends on the cat.
My three had covered litterboxes, and didn't mind it (i just switched them right to it).... when I had one covered and one not.. they prefered doing one "job" in the uncovered and one in the covered..

We're back to uncovered because Mozart is a tall kitty, and he can't maneuver very well with a lid on it (and will tend to stick his butt out the opening!!! ) (oh Gosh, Poor Mozart.. he'll never get a girlfriend if I keep telling all the girls what he does!!)

We even have big litterboxes.. can't go any bigger or Bach and Eliza won't be able to climb in!!

If Ally is picky, you might keep the old box around, but you could also just watch her. Put the new box down, take away the old, and show her the new one, and let her see there's litter in there.. then if she shows no inclination for wanting to go in there, you can switch to the old again (Or just take the cover off!)