Spin-hydrodynamic coupling is a recently discovered method to directly generate electricity from an electrically conducting fluid flow in the absence of Lorentz forces. This method relies on a collective coupling of electron spins - the internal quantum-mechanical angular momentum of the electrons - with the local vorticity of a fluid flow. In this work, we experimentally investigate the spin-hydrodynamic coupling in circular- and noncircular-capillary pipe flows and extend a previously obtained range of Reynolds numbers to smaller and larger values, 20 < Re < 21500, using the conducting liquid-metal alloy (Ga,In)Sn as the working liquid. In particular, we provide experimental evidence for the linear dependence of the generated electric voltage with respect to the bulk-flow velocity in the laminar regime of the circular pipe flow as predicted by Matsuo et al. [Phys. Rev. B. 96, 020401 (2017)]. Moreover, we show analytically that this behavior is universal in the laminar regime regardless of the cross-sectional shape of the pipe. Finally, the proposed scaling law by Takahashi et al. [Nat. Phys. 12, 52 (2016)] for the generated voltage in turbulent circular pipe flows is experimentally evaluated at Reynolds numbers higher than in previous studies. Our results verify the reliability of the proposed scaling law for Reynolds numbers up to Re = 21500 for which the flow is in a fully developed turbulent state.