Let $R$ be a complete discrete valuation ring and let $K$ be its field of fractions. Suppose $X$ is a smooth rigid-anaytic space over $K$. Often it is convenient to have a model of $X$ whose reduction has singularities which are as mild as possible--a semistable model. This amounts to having an admissible covering of $X$ by open affinoids $X_i$, each of which has good reduction, such that the reductions of any pair $X_i$ and $X_j$ meet transversally, if at all. (See the paper of Bosch/Lütkebohmert for definitions.) Let us assume such a semi-stable model of $X$ exists. Then the étale cohomology of $X$ can be computed from the combinatorics of the covering $X_i$, together with the étale cohomology of each $X_i$, via the weight spectral sequence of Rapaport-Zink.

Now suppose I have an open affinoid $Z\subset X$ which happens to have good reduction. My question is: Does there admit a semi-stable model of $X$ for which $Z$ belongs to the covering? Failing this, is there some sense one can make of my intuition that the cohomology of the reduction of $Z$ ought to contribute to the cohomology of $X$?

Feel free to edit/criticize my question to smithereens if you like.