Ken Brown (of "Brown Representability" fame) published a paper relating Cech hypercohomology and and sheaf cohomology, extending a result of Verdier (that is relegated to an appendix in, I believe, SGA 4. Brown calls it "Verdier's Hypercovering theorem"). There is an evident way to compare Cech cohomology and Cech hypercohomology (as discussed in the paper), and so I believe that it should answer your question:
Harry Gindi
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