It is well known that you can use forcing to change the truth value of various sentences ($CH$, $\Diamond$, et cetera). However, often when performing such a construction over a model $V$, the action is generally in $V[G]$ (or some inner model thereof - as when we violate $AC$ in $HOD(x)^{V[G]}$); we want to figure out what goes on in $V[G]$ for the purposes of, say, a relative consistency proof.

There are constructions, however, that also tell us about what goes on in $V$ (possibly in relation to $V[G]$), as when we consider embeddings of the form $j: V \longrightarrow M \subseteq V[G]$. Such embeddings can have quite small critical points (even as small as $\omega\_1$), giving us a new perspective on the small uncountable sets of our original model $V$.

My question:

Are there *other* forcing constructions (i.e. not generic embeddings) that tell us what the structure of our original model $V$ is like with respect to *uncountable* sets (so not in the trivial sense of showing the consistency of proof-codes in $V\_\omega$) and how $V$ relates to $V[G]$? To put it another way, are there other constructions that allow us to see the structure of $V$ more clearly from $V[G]$?