The following theorem of McAloon provides a striking analogue of your phenomenon in the realm of models of arithmetic.  The idea here is that *every* nonstandard model of the very weak theory $I\Delta_0$, which includes induction only for $\Delta_0$ formulas, has a nonstandard initial segment that is a model of PA.  Thus, this is a situation where a model of PA was continued to a model of a much weaker theory higher up. 

<b>Theorem.</b> ([McAloon, Kenneth, <i>On the complexity of models of arithmetic</i>. 
J. Symbolic Logic 47 (1982), no. 2, 403–415.](http://www.ams.org/mathscinet-getitem?mr=654796)  Every nonstandard model of $I\Delta_0$ has an initial segment that is a nonstandard model of PA.

There has been further work with these models.  For example, in [Strong initial segments of models of $I\Delta_0$, Paola D'Aquino, Julia F. Knight, Fund. Math. 195 (2007), 155-176](http://journals.impan.gov.pl/fm/Inf/195-2-4.html), the authors investigate the possibility of stronger features in the initial segments.

I am not sure to the extent to which there is a ZFC analogue of McAloon theorem.  One natural analogue of it would be the question of whether every nonstandard model of KP has an initial segment that is a model of ZFC.  I recall hearing once about this, but I don't recall now whether it was as a theorem or as a question.