@DocPop mentioned a technique to get an extra string loop around the axle using just the yoyo string, and I’ve been playing around with it, and thought I’d share a short GIF demo of the technique. I definitely can’t take credit for thinking this up, but it’s been extremely useful to me in the last few days since I lost my string pick tools and haven’t taken the time to make a new one.
Sometimes the weight of the string is enough to interfere, so you might need to hold the string in your non-yoyo hand underneath the yoyo. It seems to be a combination of a sawing motion to get the yoyo-hand string behind/underneath the extra loop around the axle, and then pulling it against the side of the yoyo.