In the usual stable homotopy category of spectra, the Postnikov truncations of any connective commutative ring is again a commutative rings, and the entire Postnikov tower can be enriched to maps of commutative rings. It follows immediately that the same result (and in particular an affirmative answer to your question) holds in the category of modules over any connective commutative ring. This is a theorem of Kriz, written up as Theorem 8.1 in Basterra's André–Quillen cohomology of commutative S-algebras. Basically, the argument is to show you can add one higher homotopy group via an extension of commutative rings by making a computation in Andre-Quillen cohomology.

In the usual stable homotopy category of spectra, the Postnikov truncations of any connective commutative ring are again commutative rings, and the entire Postnikov tower can be enriched to maps of commutative rings. It follows immediately that the same result (and in particular an affirmative answer to your question) holds in the category of modules over any connective commutative ring. This is a theorem of Kriz, written up as Theorem 8.1 in Basterra's André–Quillen cohomology of commutative S-algebras. Basically, the argument is to show you can add one higher homotopy group via an extension of commutative rings by making a computation in Andre-Quillen cohomology.