I'm thinking something along the lines of the Skis.
A brown cover around white bricks for instance will get your cover blown in a second flat. Solution? Make it like the skis, in that the cover's color depends on which paint you had last. So switch to white paint, equip the cover and there you go, nice white cover that blends in.
The cover could simply be a playerCharacter type vehicle like the horse, and your actual player could be under it. When you move around with the box it would bob up and down revealing your feet slightly.