The application of the hyperspherical adiabatic expansion to describe three-body scattering states suffers from the problem of very slow convergence. Contrary to what happens for bound states, a huge number of hyper-radial equations has to be solved, and even if done, the extraction of the scattering amplitude is problematic. In this Letter we show how to obtain accurate scattering phase shifts using the hyperspherical adiabatic expansion. To this aim two integral relations, derived from the Kohn variational principle, are used. The convergence of this procedure is as fast as for bound states.