We say that a family ${\cal S}\subseteq{\cal P}(\mathbb{N})$ is bijection-dodging if there is a bijection $\varphi:\mathbb{N}\to\mathbb{N}$ with $\varphi(T)\notin {\cal S}$ for all $T\in{\cal S}$.

Given a bijection-dodging family ${\cal S}\subseteq{\cal P}(\mathbb{N})$, is there necessarily a bijection-dodging ${\cal S}_0\subseteq{\cal P}(\mathbb{N})$ with ${\cal S}\subseteq{\cal S}_0$, and for every $X\in{\cal P}(\mathbb{N})\setminus{\cal S}_0$ we have that ${\cal S}_0\cup\{X\}$ is no longer bijection-dodging?

(Of course, one reflex is to use Zorn's Lemma, but there is a chain of bijection-dodging families with their union no longer being bijection-dodging.)