Is there a way to increase the speed of light itself?

Yes, the velocity of light can exceed $c$, but this is a somewhat technical situation and does not represent a violation of relativity.

As you say, the change in the speed of light in some material is due to an interaction of light with the electrons in that material. When you're well away from any electronic transitions the interaction is relatively small, however when the energy of the light matches an electronic transition in the material the refractive index changes rapidly and in fact can become negative.

The reason this matters is that in most materials light has two velocities, the phase velocity and the group velocity. It's the group velocity that we normally think of as the velocity of the light because the group velocity is normally the speed that information can be transmitted, and information can't be transmitted faster than light. The group velocity is given by:

$$ v_g = \frac{c}{n - \lambda \frac{dn}{d\lambda}} $$

Near an electronic transition $dn/d\lambda$ changes sign, in which case it is possible for $v_g$ to exceed $c$. This called anomalous dispersion. However under these circumstances it is no longer true that $v_g$ is the velocity that information is transmitted so there is no violation of relativity.

See this article from Nature for some nice diagrams of the refractive index change and more background info.