We have recently discovered how a novel type of hybrid improper ferroelectricity can arise in the perovskite HgMn3Mn412. This ferroelectricity is driven by a highly unusual charge transfer and orbital ordering process that induces a polar grounds state. Interestingly, isoelectronic CaMn3Mn4O12 exhibits a different sequence of phase transitions with rather different orbital physics that does not directly lead to hybrid improper ferroelectricity. The present study seeks to use high pressure to tune through the differing orbital ordered phases of HgMn3Mn3O12 giving greater insight into the physical driving forces behind this novel kind of ferroelectricity. This will help us not only gain insight into how the unusual charge and orbital physics surrounding the ferroelectric state in HgMn7O12 arises, but will give us indications how it can be induced and controlled in other related systems.