I love to spin singles on my spindles, but I’ve never been a fan of plying on a spindle.
Instead I use my Kate 45 and ply on my wheel.
All with the help of the humble drinking straw. My cop goes on the straw and the straw goes on my lazy kate rod.
I have a pack of regular bendy straws that fit perfectly over the rods on my Kate 45 just loose enough to spin, but not so loose that the fly off when plying. The straw has to be taller than your cop but, not taller than your lazy kate rod.
I slide my straw as far as it will go up the spindle shaft and slide the cop slowly onto the straw.
I found that if I tried to make the cop transfer off of the spindle or even at the bottom of the spindle shaft, it was much harder to get the cop on the straw smoothly.
I use my wheel to ply, but it’s just as easy to ply to a spindle from your kate.