This paper continues the study of the Gregory-Laflamme instability of black strings, or more precisely of the order of the transition, being either first or second order, and the critical dimension which separates the two cases. First, we describe a novel method based on the Landau-Ginzburg perspective for the thermodynamics that somewhat improves the existing techniques. Second, we generalize the computation from a circle compactification to an arbitrary torus compactification. We explain that the critical dimension cannot be lowered in this way, and moreover, in all cases studied the transition order depends only on the number of extended dimensions. We discuss the richer phase structure that appears in the torus case.