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.

Theorem. (McAloon, Kenneth, On the complexity of models of arithmetic. J. Symbolic Logic 47 (1982), no. 2, 403–415. 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, 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.

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 weak theory $I\Delta_0$, which includes induction only for $\Delta_0$ formulas, has a nonstandard initial segment that is a model of the comparatively stronger theory PA. Thus, this is a situation where a model of a strong theory, PA, was end-extended to a model of a much weaker theory higher up, and the theorem is that this is unavoidable in nonstandard models of the weak theory.

Theorem. (McAloon, Kenneth, On the complexity of models of arithmetic. J. Symbolic Logic 47 (1982), no. 2, 403–415. 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, 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.