A beautiful construction (relying on 2-dimensional hyperbolic circle packings) of hyperbolic 3-manifolds with many totally geodesic boundary components cab be found in the paper

Totally geodesic boundaries are dense in the moduli space. J. Math. Soc. Japan 49 (1997), no. 3, 589--601.

by Fujii and Soma.

A beautiful construction (relying on 2-dimensional hyperbolic circle packings) of hyperbolic 3-manifolds with many totally geodesic boundary components can be found in the paper

Totally geodesic boundaries are dense in the moduli space. J. Math. Soc. Japan 49 (1997), no. 3, 589--601.

by Fujii and Soma.