In Counting arithmetic lattices and surfaces, Mikhail Belolipetsky, Tsachik Gelander, Alex Lubotzky and Aner Shalev prove the following Theorem.

Let $H$ be a connected simple Lie group of real rank one. Then there is an effective computable constant $C=C(H)$ such that for any lattice $\Gamma < H$ we have $r(\Gamma)\le C\cdot \mathrm{vol}(\Gamma\backslash H)$, where $r(\Gamma)$ is the minimal number of generators of $\Gamma$.

Applying this to $SO(1,3)$ gives $\mathrm{vol}(M) \ge 1/C\cdot r(\pi_1(M))$ in your case.