We report new STAR measurements of midrapidity yields for the Λ, Λ[over ¯], K(S)(0), Ξ(-), Ξ[over ¯](+), Ω(-), Ω[over ¯](+) particles in Cu+Cu collisions at √S(NN)==200 GeV, and midrapidity yields for the Λ, Λ[over ¯], K(S)(0) particles in Au+Au at √S(NN)==200 GeV. We show that, at a given number of participating nucleons, the production of strange hadrons is higher in Cu+Cu collisions than in Au+Au collisions at the same center-of-mass energy. We find that aspects of the enhancement factors for all particles can be described by a parametrization based on the fraction of participants that undergo multiple collisions.