  We investigate the influence of different spin-orbit couplings on topological
phase transitions in the bilayer Kane-Mele model. We find that the competition
between intrinsic spin-orbit coupling and Rashba spin-orbit coupling can lead
to two dimensional topological metallic states with nontrivial topology. Such
phases, although having a metallic bulk, still possess edge states with
well-defined topological invariants. Specifically, we show that with preserved
time reversal symmetry, the system can exhibit a $\mathbb{Z}_2$-metallic phase
with spin helical edge states and a nontrivial $\mathbb{Z}_2$ invariant. When
time reversal symmetry is broken, a Chern metallic phase could appear with
chiral edge states and a nontrivial Chern invariant.
