Let $S$ be the unit object in a monoidal stable homotopy category $SH$ (we demand that the multiplication $S\times S\to S$ is commutative and associative on the level of spectra, and not just up to weak equivalence). Let $\tau$ be a $t$-structure for $SH$ such that $S$ is $\tau$-negative and that $SH^{\tau\le 0}\times SH^{\tau\le 0}\subset SH^{\tau\le 0}$. Then for any $m\le 0$ the object $S'=S^{\tau \ge m}$ (a 'factor' of $S$) can be easily seen to be a monoidal object in $SH$. My question is: can $S'$ be lifted to a commutative ring spectrum? Which restrictions on $SH,S,t$ are needed to do this? Are there any other ways of 'rigidifying' $S'$ such that one can still consider a certain triangulated category of $S'$-modules?

Basically I am interested in 'motivic' stable homotopy categories ($SH$, $MGL$-modules, and 'big Voevodsky's motives'), yet the easiest examples of my setting are the Postnikov $t$-truncations of the sphere spectrum in the 'usual' ('topological') $SH$ (for $m=0$ this is the Eilenberg-Maclane spectrum for $\mathbb{Z}$). Also, what can be said about the complex cobordism spectrum and modules over it?