How much cork did you leave sticking out? I just did 10 bottles with cages recently, but it's an experiment - I left as much cork sticking out as I could (1/4" max, any more and it gets stuck in the corker) and just put the cage on ontop. I'm hoping that the carbonation will push the corks out, where they will be caught by the cages.
I've tried 5 or 6 different regular floor corkers. None will allow the cork to stick out more than a 1/4" or so, any more than that and when the squeezing mechanism opens it slides to the side a little, blocking part of the hole the cork has just been shoved through. If the cork is tall enough to be sticking through that hole at all then it will be caught and cannot be removed without being ruined.
I was able to leave ~1/2" out of the top of the bottle (so that there was only about 1" between the bottom of the cork and the top of the brew)... Just enough to get the cage on, and the lower wire around the bottom rim on the neck. I didn't use the middle indent in the bottle, since I figured that would be too little cork inside the bottle.
When setting the depth (the stopper was almost all the way to the bottom of the threads on the plunger) I just needed to make sure to stop it before it went all the way down past the small lip above the jaws. I plan on making a more solid spacer before I use the corker again, so that I can get more consistent results.
Still, the little red floor corker did just fine for me. I might get the more expensive champagne corker if I use those actual bottles, and corks. From my understanding, the jaws in that one open up more, and the hole where the bottle top goes is wider.
To get the bottles out of the corker, I simply needed to use a little bit of muscle, and leverage to gently pull them down and out of the corker. Left the cork exactly as it was supposed to be, so I was able to cage it up.
I'm thinking of making something to go into my cordless drill to help wind the cage wire better. Using the screwdriver was not as much fun. Plus, I got sliced really nice (a couple of times) on the metal piece at the top of the cage. The 'tool' could be something as simple as a short piece of stiff wire, bent to hold the cage loop while I let the drill turn slowly. Should give more consistent results there. I'll let you know if/when I make that, and how it works...