We characterize the spin injection into bilayer graphene fully encapsulated in hexagonal boron nitride (hBN) including a trilayer (3L) hexagonal boron nitride (hBN) tunnel barrier. As a function of the DC bias, the differential spin injection polarization is found to rise to -60% at -250 mV DC bias voltage. We measure a DC spin polarization of similar to 50%, 30% higher compared to 2L-hBN. The large polarization is confirmed by local, two terminal spin transport measurements up to room temperature. We observe comparable differential spin injection efficiencies from Co/2L-hBN and Co/3L-hBN into graphene and conclude that the possible exchange interaction between cobalt and graphene is likely not the origin of the bias dependence. Furthermore...