  We extend a previous result [Phys. Rev. Lett. 105, 090403 (2010)] on Casimir
repulsion between a plate with a hole and a cylinder centered above it to
geometries in which the central object can no longer be treated as a point
dipole. We show through numerical calculations that as the distance between the
plate and central object decreases, there is an intermediate regime in which
the repulsive force increases dramatically. Beyond this, the force rapidly
switches over to attraction as the separation decreases further to zero, in
line with the proximity force approximation. We demonstrate that this effect
can be understood as a competition between an increased repulsion due to a
larger polarizability of the central object interacting with increased fringing
fields near the edge of the plate, and attractive forces due primarily to the
nonzero thickness of the plate. In comparison with our previous work, we find
that using the same plate geometry but replacing the single cylinder with a
ring of cylinders, or more generally an extended uniaxial conductor, the
repulsive force can be enhanced by a factor of approximately $10^3$. We
conclude that this enhancement, although quite dramatic, is still too small to
yield detectable repulsive Casimir forces.
