An inequality for sumsets

Question

Let $A \subset \mathbb{Z}/N\mathbb{Z}$ and let $m \geq 1$ be an integer such that $|A| < N^{1/m}$. I'm wondering if anyone has seen a good upper bound for the sum $$ S_m(A) = \sum_{k=1}^m|kA|, $$ where $kA := A + \cdots + A$, $k$ times. Note there is the trivial bound $S_m(A) \leq \sum_{k=1}^m|A|^k$. But can we do better?