  In this note, we prove that the syzygy bundle $M_L$ is cohomologically stable
with respect to $L$ for any ample and globally generated line bundle $L$ on an
Enriques (resp. bielliptic) surface over an algebraically closed field of
characteristic $\neq 2$ (resp. $\neq 2,3$). In particular our result on complex
Enriques surfaces improves a result of Torres-L\'opez and Zamora by removing a
condition on Clifford index. Together with the results of Camere and
Caucci--Lahoz, it implies that $M_L$ is stable with respect to $L$ for an ample
and globally generated line bundle $L$ on any smooth minimal complex projective
surface $X$ of Kodaira dimension zero.
