Multiple higher-order Stokes of rotational stimulated Raman scattering (SRS) in compressed hydrogen covered the wavelength range from ~1.1 µm to ~1.4 µm using ~1.2 ps pump pulses from a Yb:YAG laser. In this study, the influence of polarization, energy, and the focusing conditions of the pump pulse, as well as gas pressure, on the conversion efficiency and modification of the spectral envelope of rotational and vibrational SRS was investigated. The use of a supercontinuum seed, as well as circular polarization of pump pulses at high hydrogen pressure, made it possible to significantly reduce the threshold for rotational SRS and suppress vibrational Stokes modes. The cascade excitation of rotational SRS, corresponding to a shift of 587 cm−1, ensured a spectrum spanning four Stokes orders withs a conversion efficiency of 52% and an output energy exceeding 3 mJ. The synthesized spectrum corresponded to ~14 fs transform-limited pulses.