Let $H$ be a Krull monoid with finite class group $G$ such that every class contains a prime divisor (for example, a ring of integers in an algebraic number field or a holomorphy ring in an algebraic function field). The catenary degree $\mathsf{c}(H)$ of $H$ is the smallest integer $N$ with the following property: for each $a\in H$ and each pair of factorizations $z,z^{\prime }$ of $a$, there exist factorizations $z=z_{0},\dots ,z_{k}=z^{\prime }$ of $a$ such that, for each $i\in [1,k]$, $z_{i}$ arises from $z_{i-1}$ by replacing at most $N$ atoms from $z_{i-1}$ by at most $N$ new atoms. To exclude trivial cases, suppose that $|G|\geq 3$. Then the catenary degree depends only on the class group $G$ and we have $\mathsf{c}(H)\in [3,\mathsf{D}(G)]$, where $\mathsf{D}(G)$ denotes the Davenport constant of $G$. The cases when $\mathsf{c}(H)\in \{3,4,\mathsf{D}(G)\}$ have been previously characterized (see Theorem A). Based on a characterization of the catenary degree determined in the paper by Geroldinger et al. [‘The catenary degree of Krull monoids I’, J. Théor. Nombres Bordeaux23 (2011), 137–169], we determine the class groups satisfying $\mathsf{c}(H)=\mathsf{D}(G)-1$. Apart from the extremal cases mentioned, the precise value of $\mathsf{c}(H)$ is known for no further class groups.