This question is related to one of my previous question Do we have the following isomorphism for $\mathcal{Ext}$?

Let $X$ be a smooth variety (over $\mathbb{C}$) and $\Delta: X \rightarrow X \times X$ be the diagonal embedding and $p_1: X\times X\rightarrow X, ~p_2: X\times X\rightarrow X$ be the projections to the first and second components. Let $E$ be a finite dimensional vector bundle on $X$. We define $$ E_{\Delta}:=\Delta_*E $$ to be a sheaf on $X\times X$. In particular we have $\mathcal{O}_{\Delta}:=\Delta_*\mathcal{O}$. We have the sheaf Ext functor $\mathcal{Ext}$ on $X\times X$ and we can define ${Rp_1}_*\mathcal{Ext}_{X\times X}^{\bullet}(E_{\Delta},E_{\Delta})$. Sasha told me in my previous question that we have $$ {Rp_1}_*\mathcal{Ext}_{X\times X}^{\bullet}(E_{\Delta},E_{\Delta})\cong {Rp_1}_*\mathcal{Ext}_{X\times X}^{\bullet}(\mathcal{O}_{\Delta},\mathcal{O}_{\Delta})\otimes \mathcal{End}(E). $$

Now we can take the derive global section on the above (complex of) sheaves: $$ R\Gamma(X, {Rp_1}_*\mathcal{Ext}_{X\times X}^{\bullet}(E_{\Delta},E_{\Delta}))\cong R\Gamma(X, {Rp_1}_*\mathcal{Ext}_{X\times X}^{\bullet}(\mathcal{O}_{\Delta},\mathcal{O}_{\Delta})\otimes \mathcal{End}(E)). $$

Notice that when the vector bundle $E=\mathbb{C}$ the trivial bundle, we just get the Hochschild cohomology $HH^{\bullet}(X)$. So we can consider the hypercohomology $R\Gamma(X, {Rp_1}_*\mathcal{Ext}_{X\times X}^{\bullet}(E_{\Delta},E_{\Delta}))$ as a "Hochschild cohomology with coefficients in $E$" and maybe we can denote it by $HH^{\bullet}(X,E)$ (I don't know whether it has already been studied). Nevertheless I think that when $E$ is nontrivial, we do not simply get $HH^{\bullet}(X,E) \cong HH^{\bullet}(X)\otimes \text{End}(E)$.

$\textbf{My question}$ is: could we compute this $HH^{\bullet}(X,E)$? For examply, could we construct a good spectral sequence which convergence to it?