  Let $X$ be a smooth irregular $3$-fold of general type over $\mathbb{C}$. We
prove that the optimal Noether inequality $$ \mathrm{vol}(X) \ge
\frac{4}{3}p_g(X) $$ holds if $p_g(X) \ge 16$ or if $X$ has a Gorenstein
minimal model. Moreover, when $X$ attains the equality and $p_g(X) \ge 16$, its
canonical model can be explicitly described.
