In thermal dark matter models, allowing the dark matter candidate to coannihilate with another particle can considerably loosen the relic density constraints on the dark matter mass. In particular, introducing a single strongly interacting coannihilation partner in a dark matter model can bring the upper bound on the dark sector energy scale from a few TeV up to about 10 TeV. While these energies are outside the LHC reach, a large part of the parameter space for such coannihilating models can be explored by future hadron colliders. In this context, it is essential to determine whether the current bounds on dark matter simplified models also hold in non-minimal scenarios. In this paper, we study extended models that include multiple coannihilation partners. We show that the relic density bounds on the dark matter mass in these scenarios are stronger than for the minimal models in most of the parameter space and that weakening these bounds requires sizable interactions between the different species of coannihilation partners. Furthermore, we discuss how these new interactions as well as the additional particles in the models can lead to stronger collider bounds, notably in jets plus missing transverse energy searches. This study serves as a vital ingredient towards the determination of the highest possible energy scale for thermal dark matter models.