Summary: Let \(G\) be a finite abelian group. The critical number \(\text{cr}(G)\) of \(G\) is the least positive integer \(\ell\) such that every subset \(A\subseteq G\setminus\{0\}\) of cardinality at least \(\ell\) spans \(G\), i.e., every element of \(G\) can be written as a nonempty sum of distinct elements of \(A\). The exact values of the critical number have been completely determined recently for all finite abelian groups. The structure of these sets of cardinality \(\text{cr}(G)-1\) which fail to span \(G\) has also been characterized except for the case that \(|G|\) is an even number and the case that \(|G|=pq\) with \(p,q\) are primes. In this paper, we characterize these extremal subsets if \(|G|\geq 36\) is an even number, or \(|G|=pq\) with \(p,q\) are primes and \(q\geq 2p+3\).