I've had the issue. I always thought it was a poor design of the new indexed shifters, and the first click should pull twice as much cable. Friction shifters, one can just let the cable go slack, and has less problems.

I like the idea of adding the washer. I may try it.

Setting your B-Screw to get the upper jockey wheel as close to the sprockets without touching through all gears may help by minimizing chain flex.