Phase noise of the period-one (P1) nonlinear dynamical oscillation in an optically injected semiconductor laser is numerically investigated. The P1 dynamics causes the laser output intensity to oscillate at a widely tunable frequency for photonic microwave generation, although the intrinsic spontaneous emission in the laser inevitably degrades the microwave signal and manifests as the oscillation phase noise. To characterize the phase noise, the P1 microwave linewidth is first numerically examined through the rate equations with a Langevin term. The P1 microwave linewidth is found to vary with the injection parameters. It is nearly minimized when the microwave power maximizes. Owing to the laser nonlinearities, the P1 microwave linewidth can even be smaller than the free-running optical linewidth. By adding an optical feedback to the laser, the P1 microwave linewidth is found to reduce as the feedback strength and feedback delay increase, in which an inverse-square dependency is followed asymptotically. By modification to a dual-loop feedback, noisy side peaks around the central P1 frequency are effectively suppressed through the Vernier effect. The dual-loop feedback maintains a low phase noise variance over a wide tuning range of the P1 frequency, while allowing long delay times for significant P1 microwave linewidth narrowing.