In typical conditions of the observation of multiple paramagnetic Raman resonance in model heterostructures containing manganese ions, we observed a never-reported series of Raman peaks, showing up in a narrow range of applied magnetic fields. These weak "fractional" peaks are located close to the middle between the strong "integer" Mn spin-flip peaks, and they reveal a remarkably weak dependence of the intensity on the peak number, in a pronounced contrast with the "integer" series. We discuss conditions to observe the "fractional" series as well as pathways for further exploration of the effect.