  In this paper we give a different approach to determining the cardinality of
$h$-fold sumsets $hA$ when $A\subset \mathbb{Z}^d$ has $d+2$ elements. This
enables us to provide more general result with a shorter and simpler proof. We
also obtain an upper bound for the value of $|hA|$ when $A\subset \mathbb{Z}^d$
is a set of $d+3$ elements with simplicial hull.
