  We experimentally study the auto-oscillating spin-wave modes in
NiFe/$\beta-$W constriction-based spin Hall nano-oscillators as a function of
bias current, in-plane applied field strength, and azimuthal field angle, in
the low-field range of 40-80 mT. We observe two different spin-wave modes: i) a
linear-like mode confined to the minima of the internal field near the edges of
the nanoconstriction, with weak frequency dependencies on the bias current and
the applied field angle, and ii) a second, lower frequency mode that has
significantly higher threshold current and stronger frequency dependencies on
both bias current and the external field angle. Our micromagnetic modeling
qualitatively reproduces the experimental data and reveals that the second mode
is a spin-wave bullet and that the SHNO mode hops between the two modes,
resulting in a substantial increase in linewidths. In contrast to the
linear-like mode, the bullet is localized in the middle of the constriction and
shrinks with increasing bias current. Utilizing 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.
