In many interacting-electron systems, the microscopic spin angular momentum of electrons is conserved along with the macroscopic angular momentum in electrical current flows, i.e., vorticity. Such spin-vorticity coupling expands the choice of materials for spintronics devices. In this paper, we evaluate the magnitude of an alternating spin current generated by the spin-vorticity coupling with a gigahertz-order surface acoustic wave in a Cu thin film. We measure the gigahertz alternating spin current by an electrical method based on the inverse spin-Hall effect. From the amplitude of the spin current, we can determine the conversion efficiency of the angular momentum between local lattice rotation and electron spin in the Cu film. The conversion efficiency is four orders of magnitude larger than the case of spin current generation via kilohertz-order vorticity in turbulent flow of liquid mercury [M. Matsuo et al., Phys. Rev. B 96, 020401(R) (2017)]. Such a huge conversion efficiency is attributable to a smaller inconsistency of energy scale between lattice rotation and electron spin than the case of liquid vorticity.