ag.algebraic geometry - Cohomology of rigid-analytic spaces

Here's a first pass at your question; hopefully it will suggest something more definitive.

Let's imagine we were in the simplest case, where $X$ is a disk, with its smooth model
being the formal affine line over $R$, and that $Z$ was the sub-disk of elements of
absolute value less than or equal the absolute value of the uniformizer. Then we can find a semistable model in which $Z$ is one of the covering opens, by blowing up the formal affine line at the origin.

So in this test case, the answer seems to be yes .

Now in general, I think that Raynaud (and/or his collaborators or those who followed in his
tradition) will say that the open immersion $Z rightarrow X$ extends to an open immersion
of formal models. So we can blow up the smooth model of $X$ and the smooth model of $Z$
so that the latter sits inside the former. What I'm not very certain about is how much
you can control the nature of these blow-ups. (Presumably not at all in general, but you're starting in a fairly nice situation.)

Have you tried asking Brian Conrad yet?