Recently I was taught how aluminium is commercially extracted. The ore is first concentrated by leaching either by Bayer's process for red bauxite (impurity: $\ce{Fe2O3}$) or by Serpeck's process for white bauxite (impurity: $\ce{SiO2}$).
Then alumina is reduced electrolytically via Hall–Héroult's process directly to 99% pure aluminium. But for further purification to remove remaining copper or iron impurities, Hoope's process is executed to get 99.99% pure $\ce{Al}$.
Since both processes involve electrolysis, and the materials added are similar in both (except $\ce{BaF2}$) with same graphite electrodes, shouldn't we get the same purity in both process (thus not requiring a further refinement)?
Also, the arrangement of the electrodes are exactly opposite to each other. Hoope's argues that pure aluminium is lighter and thus floats so cathode is kept at top, but why doesn't that logic work in Hall–Héroult's? The difference in purity varying by 0.99% and the electrolyte used being same.
Any insight into this matter is helpful because I haven't been able to come to a convincing conclusion.