Thanks for the reference. I could be muddling things up, but isn't the conditions of no incompressible tori/annuli a condition on the manifold rather than anything to do with the hierarchy? To put it another way, suppose I have two explicitly given hierarchies $\mathcal{H}_1$ and $\mathcal{H}_2$ for a manifold $M$ that satisfies all conditions for being a hyperbolic $3$--manifold. If I cut up and re-glued each of these in turn, can it happen I end up with a hyperbolic structure on $(M,\mathcal{H}_1)$ but not on $(M,\mathcal{H}_2)$.

Yes, Having an angle structure is good enough for me. The motivation for the question is that I can construct a class of hierarchies for the manifold in question and was hoping to construct an angle structure using this (or at least to get some condition when this is possible).

Yes, I also assume that $M$ is atoroidal, thanks Neil for pointing that out.

(continued)...I guess the question is whether in this case the boundary pattern carries enough information for re-gluing. Thanks again....was just curious whether this could be done...you have more than adequately answered the original question btw:)

Thanks Ian, yes this makes sense. Just wondering whether the obstruction you mention is the only known obstruction. For instance if I have a decomposition along hierarchy surfaces that are non-separating (I can do this for example by assumptions on the first homology of the manifold) and postpone cuts by compressing disks till the end, then I think I have a construction where each hierarchy surface is non-separating. I tried this out on closed Haken $3$--manifolds with infinite homology and it seems to work. Thus I end up with a single $3$--ball and all cuts would be visible...(continued)