I still think half inch ply is the best solution. Ask an engineer. Another thin layer over a weak sub-floor does not equal the strength of the same thickness of sub-floor as when the layers are added together, unless they are incorporated by a structural adhesive. If you use a leveling compound and glue the 3/8th inch ply with a full trowel method, you may be able to get the strength of a one inch sub-floor. You will need to have complete contact between all the clean existing 5/8 ply and the adhered leveling patch. Using a half ply over the existing 5/8 ply with glue assist would be my choice.Using screws where the tarpaper shims are needed will work using the correct fastener at schedule, at a minimum.
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