Summary: The present paper is a continuation of our recent paper [J. Math. Anal. Appl. 476, No. 2, 426–463 (2019; Zbl 1516.35456)]. We will consider the following Cauchy problem for semi-linear structurally damped \( \sigma \)-evolution models: \[ u_{tt}+ (-\Delta)^\sigma u+ \mu (-\Delta)^\delta u_t = f(u, u_t), \quad u(0, x) = u_0(x), \quad u_t(0, x) = u_1(x) \] with \( \sigma \ge 1 \),\( \mu>0 \) and \( \delta \in (\frac{\sigma}{2}, \sigma] \). Our aim is to study two main models including \( \sigma \)-evolution models with structural damping \( \delta \in (\frac{\sigma}{2}, \sigma) \) and those with visco-elastic damping \( \delta = \sigma \). Here the function \( f(u, u_t) \) stands for power nonlinearities \( |u|^{p} \) and \( |u_t|^{p} \) with a given number \( p>1 \). We are interested in investigating the global (in time) existence of small data Sobolev solutions to the above semi-linear models from suitable function spaces basing on \( L^q \) spaces by assuming additional \( L^{m} \) regularity for the initial data, with \( q\in (1, \infty) \) and \( m\in [1, q) \).