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, our results show that local gating arising from the applied DC bias is not responsible for the DC bias dependence. Carrier density dependent measurements of the spin injection efficiency are discussed, where we find no significant modulation of the differential spin injection polarization. We also address the bias dependence of the injection of in-plane and out-of-plane spins and conclude that the spin injection polarization is isotropic and does not depend on the applied bias.