Addendum to "The Noether inequality for algebraic threefolds"

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 $p_g(X)\le 4$ or $p_g(X)\ge 11$, $$\text{vol}(X)\ge \frac{4}{3}{p}_g(X)-\frac{10}{3}\mathrm{.}$$