The highest-profile schemes for manipulating the speed of light to make "slow light" use quantum phenomena like electromagnetically induced transparency to manipulate the index of refraction. The "slow light" effects come not from having a large index, but rather a steep slope in the index as a function of frequency. They produce this by pumping the medium with additional light so that the combination of the two light fields puts the atoms in the medium into a particular quantum superposition. The exact final speed can be tuned by choosing the properties of the pump field appropriately.
"Fast light" experiments can work in a similar manner-- for example, this paper on superluminal pulse propagation (arxiv version) uses a cesium vapor pumped into a particular state as the medium through which the pulse propagates faster than the normal speed of light. The resulting speed depends on the parameters of the laser used to excite the vapor, and thus can be tuned by varying those parameters.
These experiments generally deal with the group velocity rather than the phase velocity, as that's the thing that you can measure in an experiment. I'm not sure if that invalidates these as responses to your question, but that's what comes to mind on the topic of varying the speed of light.