Restrictions on transitions in Spectroscopy
If the energy is to be transferred from the photon to your atom one energy gap must match the energy of the photon. Thus absorption may occur from the first to second excited state in your example as the energy in both is 2E. However, there has to be some population in the first excited state for this to happen, zero population zero absorption. In fact it is a bit more subtle in that there has to be a difference in population between the two levels involved. It is possible for the photon to stimulate a transition from the upper level to the lower one, just as from the lower one to the upper one. The reason why this seems odd is that in most cases the population in the upper level is vastly smaller than in lower ones (via Boltzmann distribution) and the population in the upper level can often be set to zero without serious error. This is particularly true in electronic spectroscopy, as in the case of atoms, but far less so for pure rotational transitions in molecules which occur in the microwave region of the spectrum. You will find details in any good spectroscopy book.

Thus neither of your cases occurs but something different; transitions between the first to second excited state. How strong this transition is depends on the population in the two levels and on selection rules involving angular momentum and electron spin changes.