Well I figured out a way to make it relatively easy to get your item back in exactly the same place, and hit all the screw holes with little, to no effort.
Here is what I do. First get some long finishing nails. I think I use 3" nails. Then once the carpet and such is back in position you use the finishing nails to poke through the carpet and insulation until you find the screw holes.
Like this...

Then once all the screw holes have been found, just lower your piece over the nails.

Then just remove one nail at a time and screw it down into place. Works perfect every time!