Assume $T$ is stable, small and $\Phi(x)$ is a formula of $L(T)$. We study the impact on $T\lceil\Phi$ of naming finitely many elements of a model of $T$. We consider the cases of $T\lceil\Phi$ which is $\omega$-stable or superstable of finite rank. In these cases we prove that if $T$ has $< 2^{\aleph_0}$ countable models and $Q = \Phi(M)$ is countable and atomic or saturated, then any good type in $S(Q)$ is $\tau$-stable. If $T\lceil\Phi$ is $\omega$-stable and (bounded, 1-based or of finite rank) with $I(T, \aleph_0) < 2^{\aleph_0}$, then we prove that every good $p \in S(Q)$ is $\tau$-stable for any countable $Q$. The proofs of these results lead to several new properties of small stable theories, particularly of types of finite weight in such theories.