|If the diaper was showing through at the top then that's why it leaked. It would wick liquid from the diaper to his clothing/bedding. Sometimes it's as easy as just adjusting the cover and making really sure the diaper (and cover tag) are tucked inside.
You mention that you just received the covers, are the diapers new as well? If so, have you prewashed them enough? They need to be prewashed at least 2-4 times (depending if they're bleached or unbleached diapers, unbleached require a little more) in order to wash out the natural cotton oils that interfere with absorbancy.
What diaper are you using for nighttime? Do you use any liners and if so, how many? Could be he needs more absorbancy...
Doesn't seem likely considering he's at the very beginning of the weight range for the size, but it could be that his build is such that a bigger cover might fit better.