Given an SFT $Σ$ and a finite set $S$ of finite words, let $Σ\langle S\rangle$ denote the subshift of $Σ$ that avoids $S$. We establish a general criterion under which we can bound the entropy perturbation $h(Σ)-h(Σ\langle S\rangle)$ from above. As an application, we prove that this entropy difference tends to zero with a sequence of such sets $S_1, S_2, \dots$ under various assumptions on the $S_i$.