Inverse zero-sum problem of finite abelian groups of rank $2$

Let $G$ be a finite abelian group and $S$ be a sequence over $G$. Let $\Sigma _k(S)$ denote the set of group elements which can be expressed as a sum of a subsequence of $S$ with length $k$. We study $\Sigma _{n^2m}(S)$ of a sequence $S$ over $C_n\oplus C_{nm}$, where $n,m$ are positive integers and $|S|=n^2m+r$ with $r\in \{nm+n-4,nm+n-3\}$. We show that either $0\in \Sigma _{n^2m}(S)$ or $|\Sigma _{n^2m}(S)|\geq (r-nm+3)nm-1$. Furthermore, we determine the structure of $S$ if $0\notin \Sigma _{n^2m}(S)$ and $|\Sigma _{n^2m}(S)|= (r-nm+3)nm-1$.