We extend the results of arXiv:1401.1645 on the generalized conformal Sp(2.n)-structure of infinite multiplets of higher-spin fields, formulated in spaces with extra tensorial directions (hyperspaces), to the description of OSp(1|2.n)-invariant infinite-dimensional higher-spin supermultiplets formulated in terms of scalar superfields on flat hyper-superspaces and on OSp(1|.n) supergroup manifolds. We find generalized superconformal transformations relating the superfields and their equations of motion in flat hyper-superspace with those on the OSp(1|.n) supermanifold. We then use these transformations to relate the two-, three- and four-point correlation functions of the scalar superfields on flat hyperspace, derived by requiring the OSp(1|2.n) invariance of the correlators, to correlation functions on the OSp(1|. n) group manifold. As a byproduct, for the simplest particular case of a conventional N=1, D=.3 superconformal theory of scalar superfields, we also derive correlation functions of component fields of the scalar supermultiplet including those of auxiliary fields.