  In this note, we continue our research on Fourier restriction for hyperbolic
surfaces, by studying local perturbations of the hyperbolic paraboloid $z=xy,$
which are of the form $z=xy+h(y),$ where $h(y)$ is a smooth function of finite
type. Our results build on previous joint work in which we have studied the
case $h(y)=y^3/3$ by means of the bilinear method. As it turns out, the
understanding of that special case becomes also crucial for the treatment of
arbitrary finite type perturbation terms $h(y).$
