Photons are elementary particles, massless, they always travel at speed c in vacuum, when measured locally.
Though, you are correct, light can slow down in media, but what actually slows down in media is the wavefront.
You are asking whether between atoms light travels at speed c or not.
Individual photons still travel at speed c in vacuum, inbetween the atoms of the media.
The "speed of light" quoted in the refractive index definition is not the speed of a photon in the material. Photons still travel at c; it's the overall (classical) light wave that seems to travel at a different speed. The microscopic mechanism of this is complicated, involving repeated absorption and re-emission of photons by the electrons in the material.
Acceleration of photon
A very simple answer: the photon is absorbed by the constituents of the medium, after which it's reemitted in the same direction (conservation of momentum). This process costs time, so the effective speed of light in the medium is reduced, while the speed of light between the absorption and the reemission stays equal to the speed of light in vacuum.
What really causes light/photons to appear slower in media?