Not all such embeddings extend, as was mentioned at Bob Lubarsky's question on Extending elementary embeddings from initial segments to all of $V$Extending elementary embeddings from initial segments to all of $V$.
Specifically, suppose that $\kappa$ is the least $1$-extendible cardinal. So there is an elementary embedding $j:V_{\kappa+1}\to V_{\eta+1}$. I claim that this embedding cannot extend to $L(V_{\kappa+1})\to L(V_{\eta+1})$, since the $j\upharpoonright V_{\kappa+1}$ is determined by $j''V_{\kappa+1}$, which is a size $2^\kappa$ subset of $V_{\eta+1}$, and by means of a flat pairing function all such subsets are coded as elements of $V_{\eta+1}$. Thus, $L(V_{\eta+1})$ would see that $\kappa$ is $1$-extendible, and so by elementarity there must be a $1$-extendible cardinal in $L(V_{\kappa+1})$ below $\kappa$, contradicting the minimality of $\kappa$.
A similar argument applies to the least $\theta+1$-extendible cardinal for any $\theta$.
