Gabai proved that homotopy hyperbolic 3-manifolds are *virtually* hyperbolic, in the paper of that name:

Gabai, David, Homotopy hyperbolic 3-manifolds are virtually

hyperbolic.

J. Amer. Math. Soc.7 (1994), no. 1, 193–198.

I suspect this is the best you can do without geometrisation.