ASM all make and sell great replacement screen covers, but they never seem to quite fit right, and no matter how careful I am, they always pop up on one side or another!

I just have to chime in and argue with this. I went through countless revisions to perfect the size of mine, and I can assure you that they will all fit perfectly. It's true they're a little bigger than the stock screen covers, but the cutting around the edge is cleaner to allow for this. There are some third party screen covers that won't sit properly because the dimensions are off, but I can assure you that isn't the case with ASM screen covers.

An unclean surface will cause lifting. If there's residue, then the adhesive isn't sticking to the plastic. Or if your backlight is too thick. The "1.5mm" backlights, which are falsely advertised and actually are closer to 2mm are too thick and may cause the shell's plastic to bend slightly.

Otherwise double sided scotch tape, you can get "craft grade" or something like that.. little stronger..

EDIT: fixed the link cause whatever converted my text to a link thought the close parentheses was part of the link..

I may just not be getting all of the old adhesive off. Any tricks for that? I've taken to scraping it, hoping that the scratched then lightly sanded surface would help adhesion, maybe that doesn't make sense for a flat, double-sided tape.

the later backlight versions basically made that bulge a non-issue.  but, the bulge problem causing the screen to pop up sounds just like what you're describing (which makes me just wonder about which version you are using).  might be good to just check and make sure the case isn't bulging still.  really....  the screens sholdn't pop up randomly unless there is a non-flat surface.  either the case, or the screen itself,

of course make sure the surface area is totally clean before application.  won't stick to dirt.

can't think of any obvious tricks to tell you.  they shouldn't be cut too large though, the screen panels are modeled from the originals and also with a case in-hand.  tight fit, yes.

i changed the adhesive on the latest batch i got, the clear ones.  if you happen to have one of those and have noticed any difference in the adhesive bond, let me know.

also, don't try super glue.  it'll eat the paint off the screen protector, or at least discolor it.

Things I've tried:

Adding rubber cement
Leaving it weighted overnight both before and after screen install, hoping the adhesive will set better
removing the thicker parts of the plastic in hopes that the backlight will sit lower
loosening screws