We experimentally study the auto-oscillating spin-wave modes in Ni(80)Fc(20)/beta-W constriction-based spin Hall nano-oscillators as a function of bias current, strength of the in-plane applied field, and azimuthal field angle in the low-field range of 40-80 mT. We observe two different spin-wave modes: (i) a linearlike mode confined to the internal field minima near the edges of the nanoconstriction, and only weakly dependent on the bias current and the applied-field angle, and (ii) a second, lower-frequency mode with significantly higher threshold current and stronger dependence on both the bias current and the externalfield angle. Micromagnetic modeling qualitatively reproduces the experimental data and reveals that the second mode is a spin-wave bullet and that the spin Hall nano-oscillator mode hops between the two modes, resulting in a substantial increase in linewidths. In contrast to the linearlike mode, the bullet is localized in the middle of the constriction and shrinks with increasing bias current. Using intrinsic frequency doubling at zero field angle, we can reach frequencies above 9 GHz in fields as low as 40 mT, which is important for the development of low-field spintronic oscillators with applications in microwave-signal generation and neuromorphic computing.