  Massive gravity theory introduced by de Rham, Gabadadze, Tolley (dRGT) is
restricted by several uniqueness theorems that protect the form of the
potential and kinetic terms, as well as the matter coupling. These restrictions
arise from the requirement that the degrees of freedom match the expectation
from Poincar\'e representations of a spin--2 field. Any modification beyond the
dRGT form is known to invalidate a constraint that the theory enjoys and revive
a dangerous sixth mode. One loophole is to exploit the effective nature of the
theory by pushing the sixth mode beyond the strong coupling scale without
completely removing it. In this paper, we search for modifications to dRGT
action by coupling the matter sector to an arbitrary metric constructed out of
the already existing degrees of freedom in the dRGT action. We formulate the
conditions that such an extension should satisfy in order to prevent the sixth
mode from contaminating the effective theory. Our approach provides a new
perspective for the "composite coupling" which emerges as the unique extension
up to four-point interactions.
