For compton scattering, electron needs to be free from any surrounding electric field. But electrons in graphite are bound to graphite. I have two doubts, first, if electrons are free in graphite, why don't they leave graphite, second if there is some force binding them, why do they show Compton scattering?
If we are doing a calculation based on the equation for the Compton effect we assume that any other interactions with the electron are negligible compared with the energy transferred between the photon and the electron. At visual wavelengths, i.e. energies of a few eV, this requires that the electron be almost completely free so it only works for an isolated electron.
However if we are doing Compton scattering with X-rays then the energies exchanged are in the keV range and vastly greater than typical electron binding energies in atoms. That means that when considering Compton scattering by X-ry photons we can treat even the electrons in atoms as though they were free.
You don't say exactly what is going on in your experiment, but if the electrons involved are the conduction electrons then the binding energy will be about the work function of graphite, which is around 4eV. For Compton scattering by any photon of an energy much greater than 4eV it is a good approximation to treat the electron as free.
When Compton scattering is exhibited the incoming photons have energies which are much larger than the binding energy of the electrons in the graphite so the electron can be considered to be "free".
Before interacting with an incoming photon the electrons do not have enough energy to escape from the graphite.
Low energy photons can give the electrons in graphite enough energy for them to escape and that is called the photoelectric effect.
Also giving the electron more energy by heating the graphite can induce thermionic emission of electrons.