If you want to find any typos, misspellings, grammatical errors, or sentences with missing words we have found the quickest way is to take a final look at the copy after the hard copy is back from the printers. Errors that eluded your notice (and those of the client, independent proof reader, and the rest of the team) are obvious under casual inspection.
This appears to be a variant of Murphy’s Law (though not the Muphry’s Law variation “if you write anything criticizing editing or proofreading, there will be a fault of some kind in what you have written”) that would go “typographical errors in your content that have eluded multiple layers of focused checking will be obvious upon casual inspection by the first uninvolved reader.”
So here’s the tip, when you ask someone to proofread something, give them a version that looks as close to the final printed form as possible (for example if it’s not fancy card stock tell them that they are looking at a Xerox copy of the final piece) and tell them “We just printed a thousand of these for next week’s event, what do you think?”
Something that I’ve found works reasonably well, especially for something you’ve seen too many times before final, is to read the piece very slowly out loud. It makes you slow down and stops your eyes from skimming over something.
as with all mehtods, not 100%, of course. Nothing beats a new set of eyes on a final.