How tall is your covered box? I use one of the "extra giant" covered boxes, which I bought at Wal-Mart, and it solved Red Cat's similar problem. I only put about two inches of litter in the box, and the box is eight inches deep up to the bottom of the opening. That solved the problem. Also, I think part of Red Cat's problem was boredom while he was an only cat. After he had another companion to fight with for the first several months, then to play with, he calmed down enough that I no longer put the covers on the boxes. Oh, the boys still fling some litter out of the box while covering their messes (my girl is much more dainty about this duty), but the amount isn't so great now as to overcome the extra work of covering and uncovering the boxes every time I scoop.
Red Cat's habit when he was an only kitty was to entertain himself by throwing a lot of litter out of the box, then running into the room from my office at one end of the room, skidding across the vinyl floor covered in litter (Wheeeeeeeeeeee!), then out the door at the opposite side of the room into the sewing room. He really just needed more active things to keep him occupied. Maybe your foster kitty does, too.