You’re right. The electrons move from one atom to the next within metals. This isn’t the case in other materials. Metals are strange.
Metals have an outer, or “conduction band”, electron that is not bound to any single atom. They move around the metal, “orbiting” everywhere. Also, electrons jump all the time. It’s just random hopping in random directions with electrons jittering everywhere, equally backwards and forwards.
An “average hopping”, also occurs during an electric current. Although electrons still jitter around, hopping between atoms and hopping in one direction, they also slowly move in one direction.
These ideas may be helpful:
Definition of an insulator: Does not contain any mobile charges. (In insulators the electrons are closely bound to the individual atoms and don’t jump between them.
Definition of conductor: A material that is filled with mobile charges.
Conductors are metals, because electrons of these metals don’t stick with individual atoms. This is the “electron gas” (or “sea charge”) of a metal in physics. It’s also known as the “metallic bonds” in chemistry.