Electrons in an atom absorbs an incoming Photon to gain energy and jump to a higher energy level (shell or different orbital) considering the fact that the photon doesn't have enough energy to make the electron jump to any higher energy level would the electron still absorb the photon?
An unperturbed atom is unlikely to absorb a photon which does not correspond to an available excited energy level for an electron.
It is unlikely because Heisenberg's uncertainty principle sets time limits on the duration of a violation of energy conservation.
Most atoms, though, are not completely unperturbed. Solids, and liquids, pressurized
gasses, molecules (many gasses are diatomic molecules), all show
fatter spectral lines than lone atoms do.
That means that there is a small possibility that an off-energy excitation can occur, according to Heisenberg, or a larger possibility when a variety of other disturbing influences are taken into consideration.
OK, the actual size of those linewidths is tiny, it isn't usually important, and only the 'right' colors of light get absorbed. But, there's enough spread that solid materials, with billions of atoms, have a fairly good chance to absorb any visible light. That's why black objects exist, and black-body approximations work.
First of all , the electron and the atom are in one quantum mechanical state. The whole atom will be aborbing the photon if there exists a difference between energy levels that, within widths, equal to the energy of the photon. If there exists, then a photon hitting an atom will turn it into an excited state, with the electron in a higher energy orbital and no photon.
If there is not fitting band energy difference, the photon will scatter off elastically on the residual electric fields of the atoms and mollecules.