I had a similar problem with my new kitten but with pee!
I highly suggest two decently large litter boxes, one for pee and one for poo!
After a week of accidents on our bed he has gone two weeks no problems since we got a second large box!
They like to use one for pee and the other for poo.
Hey! We got little Oliver about three weeks ago, he is about 11 (almost 12) weeks old and we have been raising him as an "Adventure Cat"!
He goes for walks, climbs trees, enjoys car rides and trips to the beach!
Is anyone else raising an adventure cat? :)