= Solution
Expand $L$ by a unary function symbol $f$. Let $T^*$ contain:
* every sentence of $T$;
* the <first-order axiomatization of an automorphism> saying that $f$ is bijective and preserves every function and relation of $L$;
* the sentence $\exists x\,[f(x)\ne x]$;
* for each $n\geq1$, a sentence $\theta_n$ asserting the existence of at least $n$ distinct elements.
Every finite subset $S\subseteq T^*$ contains only finitely many size requirements, say no requirement beyond $\theta_N$. By hypothesis there is a finite nonrigid model $A\models T$ with at least $N$ elements. Interpret $f$ as a nonidentity automorphism of $A$. This expanded structure satisfies $S$; indeed it satisfies every automorphism axiom, including any finitely many that occur in $S$.
Thus every finite subset of $T^*$ is satisfiable. By the <compactness theorem>, $T^*$ has a model $M^*$. The sentences $\theta_n$ make its domain infinite, while $\exists x[f(x)\ne x]$ and the automorphism axioms make $f^{M^*}$ a nonidentity automorphism. The $L$-reduct $M$ is therefore an infinite nonrigid model of $T$. This is the <compactness transfer of finite nonrigidity>.
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