However, if the Higgs field is exactly what we think it is, it seems to imply energy densities several magnitudes bigger than the currently observed vacuum energy density of the universe. This leads to either the cosmological constant is wrong and/or there was no Big Bang, or the Grand Unified Theory is missing another major component instead of merely missing the discovery of the Higgs boson and an accurate measurement of the Higgs field.
We discovered one Higgs boson that has a mass of ~126GeV, but no one is sure that it's the only one. There are plenty of alternative models that have more Higgs bosons yet to be discovered. More details in https://en.wikipedia.org/wiki/Higgs_boson#Alternative_models It's a very obscure paragraph so if you (GP) don't understand the details don't worry, me neither. But the important part is that there are many models and each one has a different number of Higgs bosons with different properties. Until other(s) Higgs bosons are discovered it's very difficult to know which model is correct.
For example there is a recent preprint about another Higgs boson with ~95GeV https://news.ycombinator.com/item?id=30807022 (88 points | 12 days ago | 26 comments) The idea is that there are two weird results in CMS and Atlas that "show" something with ~95GeV, but they only have 3 sigmas. So it may be a fluke, but it's unusual enough to keep an eye in those experiments. In the post I linked they interpret this as a family of 3 Higgs bosons, the old ~126GeV, the new dubious ~95GeV one, and a third one yet to be discovered. (Just to be super clear, the ~95GeV and the proposed family are unconfirmed.)