© 2016 American Physical Society. We report on spin transport features which are unique to high quality bilayer graphene, in the absence of magnetic contaminants and strong intervalley mixing. The time-dependent spin polarization of a propagating wave packet is computed using an efficient quantum transport method. In the limit of vanishing effects of substrate and disorder, the energy dependence of the spin lifetime is similar to monolayer graphene with an M-shaped profile and minimum value at the charge neutrality point, but with an electron-hole asymmetry fingerprint. In sharp contrast, the incorporation of substrate-induced electron-hole puddles (characteristics of supported graphene either on SiO2 or hBN) surprisingly results in a large enhancement of the low-energy spin lifetime and a lowering of its high-energy values. Such a feature, unique to the bilayer, is explained in terms of a reinforced Dyakonov-Perel mechanism at the Dirac point, whereas spin relaxation at higher energies is driven by pure dephasing effects. This suggests further electrostatic control of the spin transport length scales in graphene devices.