Rule of thumb with cats is one box per cat plus one extra. Some kitties don't like to use a box belonging to someone else. A box of his own is a must.
Can you find out what kind of litter the person who you got him from used and use the same for a while? That way it isn't a huge change from what he is used to. Also, is the box too high for him? I know you said he jumped out of the one but maybe he doesn't like having to struggle to get into it.
Don't put his new box close to the other one. If they are side by side there may be issues too. Clean the soiled spots with a cleaner that is ammonia free. You can use Nature's Miracle (expensive) or the home made recipe that one of our members shared with us.... 1/4 cup rubbing alcohol, 1/4 cup vinegar, 1 cup water mixed in a spray bottle and applied liberally to the spots kitty has soiled.
You didn't mention if the litterbox was clean or soiled when you put the kitten in it. It is imperative when the kittens are small to keep the box as clean as possible. It is also very important NOT to use clumping litter. Kittens have a habit of munching on the clay which can cause blockages in their bellies and possible death.
Assumptions do nothing but make an ass out of u and me.
We can stick our heads in the sand for only so long before it starts choking us. Face it folks. The pet population is bad ALL OVER THE WORLD!