I am trying to read the Hovey-Shipley-Smith article as defining the stable model structure on symmetric spectra as a left Bousfield localization (as explained on <a href="http://ncatlab.org/nlab/show/Bousfield+localization+of+model+categories">nLab</a>) of the projective level model structure on symmetric spectra, which has level equivalences as weak equivalences.

Almost all ingredients are there in the article. All I have left to show is that the injective Omega-spectra are indeed the S-local objects, where S is the set of level equivalences. The corollary 3.1.8 that every level equivalence induces a weak equivalence of simplicial hom-sets $Map_{Sp^\Sigma}(f,E)$ for E a injective Omega-spectrum. Conversely, lemma 3.1.5 and example 3.1.10 conspire to tell you that if a symmetric spectrum is S-local and injective, it is an Omega-spectrum. So, what remains: is any S-local symmetric spectrum injective?