I know that if we assume zero sharp we obtain for a class of standard ordinals $L_\alpha\prec L_\beta$ , $\alpha <\beta$ and obviously $L_\alpha\in L_\beta$.

My question is : If we only assume the existence of a standard transitive model ,and from it we have a countable standard ordinal $\alpha$, $L_\alpha\models ZFC$, being for example $L_\alpha$ the minimal model,then there exist $N$ elementary end-extension $L_\alpha\prec N$ and $L_\alpha\in N$?, $N$ not necessarily externally well-founded.

Any help or reference would be appreciated.