We study the bifurcation patterns of the "regularized" Kuramoto-Sivashinsky equation (RKS), which results from relaxation of the long-wave approximation in the Kuramoto-Sivashinsky equation (KS). Both equations model the flow of a viscous thin film down a vertical plane, and the RKS was recently introduced to model sharp gradient patterns that cannot be captured by the KS. We show that in some flow regimes, relaxation of the long-wave approximation, even without solution breakup, causes a profound effect on the secondary and even on the primary instabilities, which now can become subcritical.