For my cats, I bought two large ,deep litter pans, and keep them side by side in the bathroom. Prior to having the two large, I had a little one for Romeo, because I thought he would adapt better to the small one, but Sammy would use the little one- and stuff his big body and tail into the tiny box, and make a huge mess all over the bathroom, But the funny thing is, Sammy will "sweep" the litter on the floor into little piles, with his paw. He doesnt like the floor dirty I suppose! Silly kitty! It's funny to see him do it!
However, the piles are very tempting to Romeo, who will mess the piles up, and scatter the litter all over the floor again.
It's like the "Odd Couple" One roomate is neat, the other is messy!
Basically, I'm saying that all kitties are different, some are neater than others! I'm sure Ebby would adapt fine to the covered box.