**Definition (1):** A‎n‎‎ ‎inner ‎model ‎of ‎‎$‎ZFC‎$ ‎is a‎ ‎tarnsitive proper class ‎model ‎‎of $‎‎ZFC$ ‎which ‎contains ‎all ‎ordinal numbers. ‎Informally ‎we ‎denote ‎the ‎collection ‎of ‎all ‎inner ‎models ‎of ‎‎$‎ZFC‎$ ‎by ‎‎$‎‎Inn(ZFC)$.‎
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Now ‎if we ‎‎consider the partial order (reflexive and transitive) $‎\langle ‎Inn(ZFC),\subseteq ‎\rangle‎‎‎‎$ ‎with ‎end ‎points ‎‎$‎L‎$ ‎and ‎‎$‎‎V$ ‎then ‎there ‎are ‎some ‎natural ‎questions ‎about ‎other properties of this partial order:‎
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**Question (1):** ‎Is‎ $‎\langle ‎Inn(ZFC),\subseteq ‎\rangle‎$ ‎linear? ‎In the other words‎, ‎is ‎the ‎following ‎statement ‎true?‎
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‎‎$‎‎‎\forall ‎M,N\in Inn(ZFC)‎~~~M\subseteq N~~\vee~~N\subseteq M$‎
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**Question (2):** ‎Is‎ $‎\langle ‎Inn(ZFC),\subseteq ‎\rangle‎$ ‎dense?‎ ‎In the other words‎, ‎is ‎the ‎following ‎statement ‎true?‎
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$‎‎‎\forall ‎M,N\in Inn(ZFC)‎~~~(M\subsetneq N \Longrightarrow ‎\exists ‎P\in Inn(ZFC)~~~M\subsetneq P \subsetneq N‎)$‎



Is there an inner model between two distinct inner models of ZFC?
Your question is related to the concept of *degrees of constructibility*, where we say that $c\equiv\_L d$ if and only if $L[c]=L[d]$. When $c$ is well-ordered in $L[c]$, then this will be an inner model of ZFC, and so the structure theory of the degrees of construtibility are a part of your informal treatment of inner models. 


The inner models can be linear, for example, if you add a Sacks real generically over $L$, then there are precisely two inner models, $L$ and $L[s]$, because of the minimality property of the Sacks real. This also shows that the inner models may not be dense. Indeed, if one adds a Sacks real over any $V$, then there are no models between $V$ and $V[s]$, which is exactly the Sacks minimality property. Meanwhile, as Miha pointed out, it is easy to make it non-linear, and in fact many other patterns are possible.


**Theorem.** If $c$ is an $L$-generic Cohen real, then the inner models of $L[c]$ are densely ordered by inclusion, but not linearly ordered.


Proof. Every inner model $M$ with $L\subset M\subset L[c]$ is a forcing extension by a subalgebra of Cohen forcing, and all such subalgebras are themselves isomorphic to Cohen forcing. So the inner models are just of the form $L[d]$ for a Cohen real $d$, and the quotient forcing $L[d]\subset L[c]$ is still countably-dense and hence also isomorphic to Cohen forcing. But if we add a Cohen real $r$ over any model $V$, then we can let $r\_0$ be the even digits of $r$, and find $V\subsetneq V[r\_0]\subsetneq V[r]$. So we get density precisely because all the forcing extensions arise by Cohen forcing. Namely, if $L[d\_0]\subsetneq L[d\_1]$, then $L[d\_1]=L[d\_0][e]$ for some Cohen real $e$, and we may take every-other digit of $e$ to find an intermediate model between $L[d\_0]$ and $L[d\_1]$. The order is not linear, since we may split $c$ into even and odd parts, and find mutually generic reals $c\_0$ and $c\_1$, whose extensions $L[c\_0]$ nad $L[c\_1]$ are incomparable, yet have meet $L$ and join $L[c]$. QED


In answer to the related question [What can the degrees of constructibility be?](https://mathoverflow.net/q/122590/1946), I posted the following:


The article [Initial segments of the degrees of constructibility](http://link.springer.com/article/10.1007%2FBF02765036?LI=true) by Marcia Groszek and Richard Shore (Israel Journal of Mathematics
June 1988, Volume 63, Issue 2, pp 149-177) shows that 



> 
> 
> > 
> > Any constructible, constructibly countable, (dual) algebraic lattice is isomorphic to the degrees of constructibility of reals in some generic extension of L.
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> > 
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And there is a lot of further work on this topic by Groszek, Slaman and others.


