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 what 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, as I explain below.

In answer to the related question [What can the degrees of constructibility be?](http://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.

And there is a lot of further work on this topic by Groszek, Slaman and others.