It is shown that sufficiently large periodic modulations in the coefficients of a nonlinear Schrödinger equation can drastically impact the spatial shape of the Peregrine soliton solutions: they can develop multiple compression points of the same amplitude, rather than only a single one, as in the spatially homogeneous focusing nonlinear Schrödinger equation. The additional compression points are generated in pairs forming a comb-like structure. The number of additional pairs depends on the amplitude of the modulation but not on its wavelength, which controls their separation distance. The dynamics and characteristics of these generalized Peregrine soliton are analytically described in the case of a completely integrable modulation. A numerical investigation shows that their main properties persist in nonintegrable situations, where no exact analytical expression of the generalized Peregrine soliton is available. Our predictions are in good agreement with numerical findings for an interesting specific case of an experimentally realizable periodically dispersion modulated photonic crystal fiber. Our results therefore pave the way for the experimental control and manipulation of the formation of generalized Peregrine rogue waves in the wide class of physical systems modeled by the nonlinear Schrödinger equation.