Seems that $S = O_{2n}^\pm(2)$ are examples of this for $n=5$, and probably for all $n \geq 5$. Such $S$ is a Jordan-Hölder factor of the automorphism group of the extraspecial group $2^{1+2n}_\pm$, so $d_2 \leq 2^n$. But the Schur multiplier is trivial, so $d_0 = d_1$, and the ATLAS of Conway et al. reports minimal faithful representations of dimensions $154$ for $O_{10}^-(2)$ and $155$ fir $O_{10}^+(2)$, both larger than $2^5 = 32$.

Seems that $S = O_{2n}^\pm(2)$ are examples of this for $n=5$, and probably for all $n \geq 5$. Such $S$ is a Jordan-Hölder factor of the automorphism group of the extraspecial group $2^{1+2n}_\pm$, so $d_2 \leq 2^n$. But the Schur multiplier is trivial, so $d_0 = d_1$, and the ATLAS of Conway et al. reports minimal faithful representations of dimensions $154$ for $O_{10}^-(2)$ and $155$ for $O_{10}^+(2)$, both larger than $2^5 = 32$.