You asked about positive instances, so here is a natural one.

**Theorem.** If $M\models\text{PA}$ is nonstandard, and $S< R$ are cuts in $M$ with $R\prec M$, then the double overspill principle holds. 

Proof. Assume $S<R\prec M$, and suppose $\varphi(s,r)$ holds in $M$ for all $s\in S$ and $r\in R$. Since by elementarity $R$ sees that $\varphi(s,r)$ holds for each particular $s\in S$, and since $S$ cannot be definable in $R$, as it satisfies $\text{PA}$, it must be that there is some $s_0\in R-S$ with $\varphi(s_0,r)$ for all $r\in R$. But now, $M$ sees all those facts, and since $M$ cannot define the cut determined by $R$, it must be that $\varphi(s_0,r_0)$ for some $r_0\in R$. So we've achieved the double overspill principle. QED 

Indeed, we achieved a stronger principle: if $S<R\prec M$ and $\varphi(s,r)$ for all $s\in S, r\in R$, then there is $s_0\in R-S$ and $r_0>R$ with $\varphi(s_0,r_0)$. By applying ordinary overspill to $R\prec M$, one can also get $s_0>R$ if desired.