Summary: In the paper [the authors, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 121, 82–100 (2015; Zbl 1320.35215)] we explained that it is reasonable to divide the study of global existence of small data solutions to semi-linear classical damped wave models with time-dependent speed of propagation, time-dependent dissipation and power nonlinearity into two cases. The super-exponential case was treated in [loc. cit.]. The present paper is devoted to the sub-exponential case. Both cases arise from a different influence of the interplay of time-dependent coefficients on the critical exponent. Here we only sketch differences to the approach for the super-exponential case. These differences appear in handling the nonlinearity. The corresponding Matsumura type estimates for a family of linear Cauchy problems depending on a parameter coincide in both cases (see [the authors, in: Fourier analysis. Pseudo-differential operators, time-frequency analysis and partial differential equation. Based on the presentations at the international conference, Aalto University. Cham: Birkhäuser/Springer. 9–45 (2014; Zbl 1332.35038); loc. cit.]).