I couldn't find any info on what set of compact objects generates the following subcategory:

Let $k$ be a field of positive characteristic and let $G$ be either a finite group or a finite group scheme over $k$. Then let $\mathrm{stab}(k[G])$ be a stable model category consiting of finitely generated $k[G]$-modules (modulo the projectives). It is a subcategory of $\mathrm{Stab}(k[G])$ - the stable model category of all $k[G]$-modules (modulo the projectives). For $\mathrm{Stab}(k[G])$, it is known that compact objects are precisely finitely-generated modules and the simple modules generate $\mathrm{Stab}(k[G])$.

More generally, let $\mathcal{K}$ be a compactly generated stable model category with a set $\mathcal{C}$ of compact objects and a set $\mathcal{G}$ of compact generators. What are the compact generators of the subcategory of $\mathcal{K}$ spanned by objects in $\mathcal{C}$? I realize that this may be unknown, so I'd appreciate any particular examples (I'm more interested in cases where the orignal stable model category has a set of compact generators rather than a single one).

For the definitions I use, I refer to Schwede-Shipley. Of course, substituting the term "stable model category" for "triangulated category" across my question makes no difference.