Spent way too long fighting the height problem on my retro motherboard project. The emulator boards I built sit where the original chips go, but with standard header strips the whole thing stood proud of the socket and the lid would not close. Nearly started snipping plastic spacers off with side cutters, wich felt like a great way to wreck a board. Then I tried machine pin header strip with no spacer at all. Pop the plastic off, snap the strip into short sections, and solder just the short stub through the pad. Round pins, barely any height, and they actually seat in the socket nicer than the chunky square ones did.
Takes a bit of care keeping them vertical while the joint sets. I tack one pin, check it sits square, then do the rest. Whole board now clears the case with room to spare. Should have done this from the start instead of stacking adapters and hoping.