In some of my recent reading I've come across a fascinating possibility for exotic hadrons that hasn't gotten enough attention. But, I don't have time at the moment to fully discuss it.
The core idea is that there a hadron states that are not inconsistent with the Standard Model QCD Lagrangian which are inconsistent with the "constituent quark model". In other words, there might be some hadrons that are neither made up of a specific combination of an integer number of quarks, nor a linear combination of such integer numbers of quarks, and that are also not glueballs. Thus, some hadrons might not be possible to describe either in terms of quark components, or as glueballs. Or, at any rate, might not have a straight forward connection to their quark components, for example in the case of hypothetical hybrid mesons (see also Section 14.3 of this PDG review).
There are no hadron resonances definitively identified with such states, although there are some suspected resonances that might fit this explanation which are being studied further at this point.
While this isn't strictly speaking "beyond the Standard Model" new physics since it simply draws on existing, consensus QCD equations, it would be a huge paradigm shift in terms of what kinds of exotic particles are possible in the universe that is almost completely unexplored so far.
I'll post more when I have it doped out better and I am in a better position to cite to a coherent set of scholarly papers on the subject. For example, this power point seems to clarify that the constituent quark model is really just a low energy simplification of QCD that omits the impact of gluons and "sea quark-antiquark pairs." These arise in more sophisticated "parton" models.