The modern mind picture of electrons bound to a nucleus is that of an orbital rather than of an orbit as you seem to be thinking.
However, your ideas are still somewhat meaningful in this modern picture in that region of electron delocalisation is more tightly confined around the nucleus for lower energy orbitals than it is for higher energy ones. You might say higher energy orbitals are bigger than lower energy ones.
Given the correspondence between the orbital and orbit picture I've just spoken of, think of your problem in terms of orbits of satellites and the energy needed to change an orbit. You need to put energy into a satellite to raise it to a bigger radius orbit. That is, to raise it to an orbit wherein the gravitational potential energy is higher. However, Earth's gravity gets weaker the further you go out into space. Think of it in these terms and I think you'll see that your reasoning is the wrong way around.
The pertinent quantity in determining the energy it takes to get to an orbit is not the strength of the gravitational field at that orbit, but the gravitational potential energy of that orbit relative to those around it.