  In this paper, we study the instability of highly-oscillating solutions to
semi-linear hyperbolic systems. A instability criterion was given in \cite{Lu}
under rather strong separation conditions of resonance sets: coupled resonance
sets are pairwise disjoint. Here we show that such separation conditions in
\cite{Lu} can be relaxed: one of the coupled non-transparent resonance sets is
allowed to intersect with at most two others. We obtain the same instability
criterion as in \cite{Lu}. Finally, we give some applications to coupled
Klein-Gordon systems with equal masses and nonlinear terms specified
particularly, where on the intersections of resonance sets, the related
interaction coefficients are non-transparent.
