Photonic microwave generation based on period-one (P1) dynamics of an optically pumped spin-polarized vertical-cavity surface-emitting laser (spin-VCSEL) is investigated numerically. Here, the frequency tunability of the photonic microwave generated from a free-running spin-VCSEL is demonstrated. The results show that the frequency of the photonic microwave signals can be widely tuned (from several gigahertz to hundreds of gigahertz) by changing the birefringence. Furthermore, the frequency of the photonic microwave can be modestly adjusted by introducing an axial magnetic field, although it degrades the microwave linewidth in the edge of Hopf bifurcation. To improve the quality of the photonic microwave, an optical feedback technique is employed in a spin-VCSEL. Under the scenario of single-loop feedback, the microwave linewidth is decreased by enhancing the feedback strength and/or delay time, whereas the phase noise oscillation increases with the increase of the feedback delay time. By adding the dual-loop feedback, the Vernier effect can effectively suppress the side peaks around the central frequency of P1, and simultaneously supports P1 linewidth narrowing and phase noise minimization at long times.