Framed Litter Box Problems

We have a framed litterbox for our two cats. We also have a mat underneath it to prevent litter from getting all over the floor.

Tonight, one of the cats dug so hard to cover their poo that they actually flung litter and small pieces of the poo all over the floor! And then of course tracked the litter clear down the hallway.

Will getting a completely covered litterbox help with this problem? It's not the first time this has happened.

Either a covered litter box or a larger box will work. I use the big plastic bins that slide under you bed and you store sweaters in. Some people use big plastic bins with holes cut in the side so the cat can go in and out, and the litter will stay in place.