The Čech-to-cohomology spectral sequence is fundamental in proving foundational results on cohomology of sheaves, and is not only used for Zariski coverings.
If $f: X \rightarrow Y$ is a finite morphism between smooth varieties of same dimension over an algebraic closed field, then $f$ is flat by miracle flatness, hence a fppf covering. As I am interested in the relations of Picard groups, when will the spectral sequence for fppf cohomology of $\mathbb G_m$ degenerate at $E_2$ page? For example, how about the smooth projective curve case which I am mostly interested in? If it does not degenerate, can we at least say something in special cases?