Since you're a puck user, I have a hard time believing your CYA is as low as you say it is. Sometimes the CYA test can be tricky. One thing you could do as a sanity check is to perform a diluted CYA test - take a sample of pool water and dilute with distilled water in a 1:1 ratio (50% dilution). Then perform the CYA test and multiply the results by 2.

So you turned up you chlorinator and now your CCs are up to 0.6ppm? That's a bit troubling to me. You could be on the verge of an algae bloom. I find it very disturbing when pool builders say "if the water is clear then it's fine." That's a hugely misleading and false statement. Clarity is not an indication of sanitary water.

As for CYA, I prefer to use solid stabilizer (100% CYA) sometimes called conditioner, and then I put it in a nylon skimmer sock and let it sit in front of a pool return until it dissolves. It can take up to a week for the added CYA to register on the CYA test so it's best to raise CYA in small increments. Since you use picks for chlorination, your CYA should naturally rise in its own unless you do lots of major water exchanges.

Retest the CYA using the dilution method and post results.