Recently, acoustic vortices with orbital angular momentum (OAM) have attracted great attention and interest owing to their extensive applications. Based on the generalized Snell’s law (GSL), acoustic vortex beams can be observed by constructing different types of acoustic metasurfaces immersed into cylindrical waveguides. However, owing to the restrictions of the phase gradient of metasurface structures and the critical order of propagating vortex beams, there exist great challenges in realizing complex modulation of vortex beams and designing advanced multifunctional sound devices. Here, we theoretically study and experimentally demonstrate the extended GSL in designing a mode converter of vortex beams based on phase-gradient acoustic metagratings (AMs). We demonstrate that the extended GSL can not only accurately predict the transmitted or reflected orders of vortex beams created by a series of AMs with arbitrary phase gradient ξ, but also effectively overcome the limitations of both the phase gradient and the critical order. More importantly, we design a type of mode converter of vortex beams and realize the mode conversion of output vortex beams by simply modulating the orders of input vortex beams. To present it, we experimentally demonstrate the switch between the output vortex beams with the orders of −2 and 2 through the same phase-gradient AM by using the input vortex beams with the orders of 1 and −1, respectively, which can be separately predicted by the GSL and the extended GSL. Our work paves a way for the complex modulation of acoustic vortex beams and the design of multifunctional OAM-based sound devices with practical applications.