We extend the result of Kowalczyk, Martel, and Mu & ntilde;oz [J. Eur. Math. Soc. (JEMS), 24 (2022), pp. 2133-2167] on the existence, in the context of spatially even solutions, of asymptotic stability on a center hypersurface at the soliton of the defocusing power nonlinear Klein-Gordon equation with p>3 , to the case 2 >= p>(5)/(3) . The result is attained performing new and refined estimates that allow us to close the argument for power law in the range 2 >= p>(5)/(3) .
