Let $\Gamma$ be a torsion-free lattice in $SO_0(3,1)$, and let $M = \Gamma \backslash {\bf H}^3$ be the corresponding hyperbolic $3$-manifold. It is well-known that in the presence of a closed, embedded, totally-geodesic surface in $M$, the canonical flat conformal structure on $M$ can be deformed via the bending construction. Equivalently, the lattice $\Gamma$ admits non-trivial deformations into $SO_0(4,1)$. We present a new construction of infinitesimal deformations for the hyperbolic Fibonacci manifolds, the smallest of which is non-Haken and contains no immersed totally geodesic surface.