  We study a previously largely unexplored branch of homogeneous and isotropic
background solutions in ghost-free massive bigravity with consistent double
matter coupling. For a certain family of parameters we find `self-inflated'
FLRW cosmologies, i.e. solutions with an accelerated early-time period during
the radiation-dominated era. In addition, these solutions also display an
accelerated late-time period closely mimicking GR with a cosmological constant.
Interestingly, within this family, the particular case of $\beta_1=\beta_3=0$
gives bouncing cosmologies, where there is an infinite contracting past, a
non-zero minimum value of the scale factor at the bounce, and an infinite
expanding future.
