Let $\mathcal{T}$ be a triangulated category and take $\mathcal{S}$ a triangulated subcategory. Consider the Verdier quotient $\mathcal{T} \left/ \mathcal{S} \right.$, morphisms in this category are constructed as roofs $E \leftarrow H \rightarrow F$ where $\text{cone}(H \rightarrow E) \in \mathcal{S}$ modulo a certain equivalence relation. In general it's not easy to compute morphisms in a general Verdier quotient because one needs to replace either E or F with a "resolution" in terms of objects which are in the left, respectively right, orthogonal to $\mathcal{S}$. An example to keep in mind is the derived category of an abelian category, in which case we replace $E$ with a projective resolution or $F$ with an injective resolution. However, in the case $D(\mathcal{A})$, where $\mathcal{A}$ is an abelian category, one has some spectral sequences converging to Homs in $D(\mathcal{A})$ (I'm thinking of $Hom_{D(\mathcal{A})} ( E, H^q(F)[p]) \implies Hom_{D(\mathcal{A})}(E,F[p+q])$ for example).

My question is whether there exist tools like such (spectral sequences or any kind of approximation) to compute morphisms in $\mathcal{T} / \mathcal{S}$. Even a way to relate them to morphisms in $\mathcal{T}$ would be fine. I hardly think such a thing exists in general, but I am mainly interested in the case $\mathcal{T} = D(\mathcal{A})$.

Thank you in advance for the help.