Does the design of the new collider vary according to which particles they'll want to probe, or is it one-size-fits-all?

Well, if a super collider is built, the LHC will be used as a pre-accelerator, as shown on the map presented in the post. The magnets will have to be all replaced, and I have no idea what will happen with the former ones. Maybe will you be able to buy one for your living room ;)

Does the design of the new collider vary according to which particles they'll want to probe, or is it one-size-fits-all?

We don't know what we are exactly looking for, so that we must stay open to any option. Some of them will be reachable with the new collider, some of them not. But at some point, a line must be drawn. With the energy requirement of the next machine, we have some certainty to learn a lot about the Higgs. This is already something. For the rest, it really depends how evilistic is nature.

Yes! Two found it here :)