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.