NOTE: expert Josh Marlan has noted that too much pressure on the sides of the saya can cause it to split, and another has said he has pushed the tape deeper into the saya. We have never encountered these problems, but be forewarned.
My swords are shimmed with naturally-sourced, renewable horn alternative. I use cut off fingernails from someone who keeps them long. Cut a small piece, glue it on, file it down. Voila, tight fit.
Hence the "quick and dirty" comment, I suppose. My concern would be more in line with the ramifications of putting excessive pressure at a right angle to the glue seams, as saya shims are usually placed at the ha and mune locations, so that the pressure is not directly working against the adhesive that's keeping the saya together.
This is so basic, that i am confused so many people thanks you for this tip. I recommend more than this felt tape veneer/ply (idk which english word is correct, i mean small thin parts of wood, thin almost like paper, which is secondary products on lumbermills or from carpenter). If after repair is koiguchi too tight, u can reduct it with filing. And i recommend to have it enought tight that part of habaki is visible and sword isnt "to the limit" in saya.