I assume that by "high-level waste" you are referring primarily to spent reactor fuel.
The precise stages of used fuel management depend on what the long-term plan for the fuel is. There are actually a lot of very useful things you can do with spent nuclear fuel; this includes recycling and reintroducing it back into the fuel cycle where even more energy can be produced, as well as the extraction of critical medical isotopes. Reprocessing has the added benefit of greatly reducing the volume of final waste product that will need to be disposed of.
Many of these separation and reprocessing techniques require that the waste first be dissolved in solution. Storing the waste in its original solid state is thus "dry storage", as opposed to storage in solution or liquid state (long-term storage of liquids saturated with radioactive heavy metals is inadvisable, for obvious reasons).
Even if you have a very efficient fuel cycle that consumes a high percentage of this high-level waste, you will inevitably have some amount of left-over waste that can't be reused. At this point, the only real option is geological repositories--you have to bury it deep in the earth. The article you referenced provides an excellent review of various approaches to this final disposal problem. Often, before final storage, the waste will undergo a final treatment to reduce volume, stabilize the solid, and discourage leaching.
In short, the answer to your question is that the stage after dry storage is...more storage--underground, for a very, very long time (think hundreds to thousands of years). The actual volume of high-level waste that requires deep geologic disposal is still quite small, but it does present a technical challenge. In my opinion, it's a manageable problem.
Hope that helps.
