  In this short note, we give a refinement of our previous work
(arXiv:1803.05553) stating that for a projective $3$-fold $X$ of general type
with either $p_g(X)\leq 4$ or $p_g(X)\geq 11$, $$\text{vol}(X)\geq
\frac{4}{3}p_g(X)-{\frac{10}{3}}.$$
