Don't take anything apart. Don't spray WD40 in there unless you want to ruin your clutch.
Try a return spring on the clutch fork. The stock adjuster puts the TOB into constant contact with the pressure plate fingers. The aftermarket adjusters don't, so the fork is free to flop back & forth bouncing the TOB off of the pressure plate giving you 'chirping.' Giving it more slack doesn't help, there's nothing pulling the fork away. Tightening it up (or putting gentle pressure on the pedal) will stop the chirp but puts you back into constant contact with the plate fingers, making your TOB wear out much more quickly. Anyone who has an aftermarket adjuster without a return spring who claims they don't have chirping has adjusted their TOB to where it's touching the pressure plate at rest.
They used to sell one called the "LDC Chicago Clutch Freeplay Correction Kit." Can't find those any longer but you could easily make your own. That silenced my chirping instantly, took up the dead-spot on top of the pedal motion, keeps the TOB from wearing out prematurely, and took all of 10 minutes to install.
If you think about our linkages, every aftermarket adjuster should come with this setup (look at any old cable-driven non-self-adjusting clutch linkage and you'll see a return spring). The chirping will quickly destroy a TOB so don't let it go too long.