This paper develops the investigations started in a work by Akbarov, Kepceler and Mert Egilmez (2011) and studies the some particularities of the torsional wave dispersion in a finitely pre-deformed sandwich hollow cylinder. The mentioned particularities relate to the influence of the stiffness ratio of the core and face layers' materials and the influence of the thickness ratio of these layers on the dispersion character of the wave under consideration. Moreover, the mentioned influences are studied for the various values of the parameter which characterizes the initial strains in the cylinder's layers. The mechanical relations of the materials of the cylinders are described through the harmonic potential. The analytical expression is presented for the low wave number limit values of the torsional wave propagation velocity. The numerical results on the foregoing influences are presented and discussed.