Čech lifting below the first possible local cohomology degree ($0\to {\check{H}}^i(\mathcal{U},\mathcal{F})\to H^i(X,\mathcal{F})\to {\prod }_a{H}^i(U_a,\mathcal{F})$)

For a finite cover whose nonempty intersections have zero sheaf cohomology in degrees $1,\dots ,i-1$, the displayed sequence is exact for $i>0$. Apply the Čech cochain complex to a flasque resolution. Local primitives and successive primitives of their overlap differences reduce a class with zero restrictions to a Čech cocycle with values in the original sheaf. Vanishing in degree $i$ on all intersections is not needed.