We have the Adams SS with $$ E_2^{p,q} = Ext^{p,q} _{E^*(E)}([S,E],[S,E]) $$ where $E$ is the Eilenberg-Maclane Spectrum yielding $\mathbb{Z}/p$ coefficients.

I was wondering if there is a SS for arbitrary compactly generated triangulated categories of which this is special case.

More specifically I am curious if we assume our category to have enough projectives (or injectives) can we avoid invoking the smash product? I am new to Stable Homotopy theory and think of smash products as black box. It would be really delightful if I could replace having a smash product by having enough projectives or something similar but algebraic.