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?, I posted the following:
The article Initial segments of the degrees of constructibility 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.
And there is a lot of further work on this topic by Groszek, Slaman and others.