First-order reversal curve (FORC) analysis allows one to investigate composite magnetic materials by decomposing the magnetic response of a whole sample into individual responses of the elementary objects comprising the sample. In this work, we apply this technique to analysing silicone elastomer composites reinforced with ferromagnetic microparticles possessing low intrinsic coercivity. Even though the material of such particles does not demonstrate significant magnetic hysteresis, the soft matrix of the elastomers allows for the translational mobility of the particles and enables their magnetomechanical hysteresis which renders into a wasp-waisted major magnetization loop of the whole sample. It is demonstrated that the FORC diagrams of the composites contain characteristic wing features arising from the collective hysteretic magnetization of the magnetically soft (MS) particles. The influence of the matrix elasticity and particle concentration on the shape of the wing feature is investigated, and an approach to interpreting experimental FORC diagrams of the MS magnetoactive elastomers is proposed. The experimental data are in qualitative agreement with the results of the simulation of the particle magnetization process obtained using a model comprised of two MS particles embedded in an elastic environment.