We present results for the matrix elements of the additional ∆S = 2 operators that appear in models of physics beyond the Standard Model (BSM), expressed in terms of four BSM B-parameters. Combined with experimental results for ∆MK and K , these constrain the parameters of BSM models. We use improved staggered fermions, with valence HYP-smeared quarks and N f = 2 + 1 flavors of "asqtad" sea quarks. The configurations have been generated by the MILC collaboration. The matching between lattice and continuum four-fermion operators and bilinears is done perturbatively at one-loop order. We use three lattice spacings for the continuum extrapolation: a ≈ 0.09, 0.06 and 0.045 fm. Valence light-quark masses range down to ≈ m phys s /13 while the light sea-quark masses range down to ≈ m phys s /20. Compared to our previous published work, we have added four additional lattice ensembles, leading to better controlled extrapolations in the lattice spacing and sea-quark masses. We report final results for two renormalization scales, µ = 2 GeV and 3 GeV, and compare them to those obtained by other collaborations. Agreement is found for two of the four BSM B-parameters (B2 and B
SUSY 3). The other two (B4 and B5) differ significantly from those obtained using RI-MOM renormalization as an intermediate scheme, but are in agreement with recent preliminary results obtained by the RBC-UKQCD collaboration using RI-SMOM intermediate schemes.