We study the current tunable microwave signal properties of nano-constriction-based spin Hall nano-oscillators 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 Jth = 1.7 x 10(8) A/cm(2). While the current dependence of the microwave frequency shows the same generic non-monotonic behavior for all w >= 80 nm, the quality of the generated microwave signal improves strongly with w, as the total power increases and the linewidth decreases linearly with w. As a consequence, the peak power for a 140 nm nano-constriction is about an order of magnitude higher than that for an 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.