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To clarify (sorry!), when I am doing the second step of screwing the additional plywood into the subfloor, should I be adding screws all over the place (i.e. where there are no joists) or only into the joists like I did in step 1 before adding the additional ply?
You re-nail/screw the subfloor to the joist before installing an underlayment. The 3/8 plywood UL is installed by spacing the fasteners every 6"-8" on center on the perimeter and about 8"-10" in the field. Don't worry about hitting the joist here. Leave an 1/8" gap between each sheet.
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I take it then, then I am going to have to add an addition layer of plywood over this?
This will have to go perpendicular to the first layer correct? 3/8" is how thick I want to go?
You will need a minimum layer of 3/8" if your subfloor is 5/8" OSB. The strongest way to run an underlayment is crossing the joist, same as the subfloor. But since you are using a 3/4" solid product either way you want to run it is OK. Keep the UL seams offset and staggered from the subfloor seams.