  Let $H$ be a transfer Krull monoid over a subset $G_0$ of an abelian group
$G$ with finite exponent. Then every non-unit $a\in H$ can be written as a
finite product of atoms, say $a=u_1 \cdot \ldots \cdot u_k$. The set $\mathsf
L(a)$ of all possible factorization lengths $k$ is called the set of lengths of
$a$, and $H$ is said to be half-factorial if $|\mathsf L(a)|=1$ for all $a\in
H$. We show that, if $a \in H$ and $|\mathsf L(a^{\lfloor (3\exp(G) - 3)/2
\rfloor})| = 1$, then the smallest divisor-closed submonoid of $H$ containing
$a$ is half-factorial. In addition, we prove that, if $G_0$ is finite and
$|\mathsf L(\prod_{g\in G_0}g^{2\mathsf{ord}(g)})|=1$, then $H$ is
half-factorial.
