Just out of interest, why do you believe this to be true? There existed particle colliders before the LHC, none were able to experimentally verify the Higgs.
It's unclear to me if it could have ever pushed this to enough sigmas to claim discovery.
Your statement that "it certainly won't be with a larger collider, especially since the last search failed" frankly makes no sense, and the claim that "the higgs was already found before" is incorrect.
This is mostly true though: "the main thing behind the LHC was to see SUSY or general new physics beyond the SM" (I say "mostly" because the energy range was chosen to ensure that either the Higgs or something else would be found; if it existed, it had to be in that range, and if not, something else had to do its job).
And that statement remains true about the next collider. It will look for "SUSY or general new physics beyond the SM".
Finally, another comment said the LHC was to study the higgs. I'm not sure about that since the higgs was already found before, it just wasn't at the significance level particle physicists like. Most of the people I talked who worked on the LHC told me the main thing behind the LHC was to see SUSY or general new physics beyond the SM.