Spin transport phenomena have been shown to be highly enhanced when the temperature approaches the Curie point of the material sustaining a spin flow. Here we propose a simple - yet unifying - explanation for such enhancements, based on a random-phase model accounting for the spin fluctuations within a ferromagnetic material in the paramagnetic phase. We show that pure spin currents carried by conduction electrons injected into a paramagnetic lattice of mutually interacting localized magnetic moments can be enhanced close to the Curie temperature by the exchange interaction between the lattice sites and the non vanishing spin density associated with the spin current. The latter partially aligns the magnetic moments of the lattice, generating a flow of paramagnons that contribute to the total spin current, resulting in an enhancement that can be as large as tenfold.