I have read the following.
Proposition. Let $T$ denote the first-order theory of $\mathbb{N}$ in the language of arithmetic. Then any countable non-standard model of $T$ is order isomorphic to a copy of $\mathbb{N}$ followed by $\mathbb{Q}$-many copies of $\mathbb{Z}$.
Now assume there exists an inaccessible cardinal, call it $\kappa$, and let $T$ denote the first-order theory of $(V_\kappa,\in)$. Let $C$ denote a non-well-founded $T$-model that is countable, and $U$ denote a non-well-founded $T$-model having cardinality $\kappa,$ so that in particular it is the same cardinality as $V_\kappa.$
Can we make any statements about the order-theoretic structure of $\mathrm{On}^C$ and/or $\mathrm{On}^U$?
The order types of ill-founded models were described, to an extent, by Friedman.
He mimics the analysis of PA models using Cantor normal forms. There are essentially two cases, depending on whether or not $M$ is an $\omega$-model. If it is then the Cantor normal forms are relatively well behaved; you can partition the ordinals of $M$, as in the PA case, into blocks of length the well-founded height of $M$ and these blocks form a dense linear order. So in the case that $M$ is an $\omega$-model of well-founded height $\alpha$, the ordinals of $M$ have order type $\alpha\cdot(1+\eta)$ where $\eta$ is some dense linear order. In particular, if $M$ is countable this is just $\alpha\cdot (1+\mathbb{Q})$.
If $M$ isn't an $\omega$-model the blocks we used above might not all have the same length and things get more complicated. Nevertheless, Friedman gives something of a description of the order type here as well. In particular, if $M$ is countable then the order type is determined by the standard system of $M$, i.e. the collection of intersections of elements of $M$ with the (collapse of the) well-founded part of $M$.
Saying anything more that this for structures of higher cardinality would require saying more about dense linear orders of that cardinality. Hutchinson has managed to obtain nice descriptions for $\aleph_1$-like models, but I doubt much more is known since the question of order types in higher cardinalities is a major open problem even for models of PA, let alone set theory.