  We study the current tunable microwave signal properties of nano-constriction
based spin Hall nano-oscillators (SHNOs) in oblique magnetic fields as a
function of the nano-constriction width, $w=$~50--140 nm. The threshold current
is found to scale linearly with $w$, defining a constant threshold current
density of $J_{th}=$ 1.7 $\times$ 10$^{8}$ A/cm$^2$. While the current
dependence of the microwave frequency shows the same generic non-monotonic
behavior for all $w\geqslant$ 80 nm, the quality of the generated microwave
signal improves strongly with $w$, showing a linear $w$ dependence for both the
total power and the linewidth. As a consequence, the peak power for a 140 nm
nano-constriction is about an order of magnitude higher than that of a 80 nm
nano-constriction. The smallest nano-constriction, $w=$ 50 nm, exhibits a
different behavior with a higher power and a worse linewidth indicating a
crossover into a qualitatively different narrow-constriction regime.
