Charles, I'm not sure of your meaning of "tool through the stabilizer hole". If you are talking about getting the tool out of the Stabilizer accessory, I have always taken the tool out by sliding the part of the tool mounted in the handle out by moving the tool toward the head stock. It can tick a little when removing the tool but tiny twists one way or the other is all it takes to get it out.
If you are talking about getting the tool out of the hollow form, it's a matter of being careful when you remove the tool to avoid damaging the lip of the piece. You can stop the lathe an remove it, but generally it just carefully remove it while the lathe is running. Same goes for sliding it in. There are, of course, some limits as to how narrow the piece opening is depending on which of the tools you have (5/8" or 3/4")
I'm not sure about an "emergency". I don't believe I've ever had serious catches. One thing to do to avoid catches is angle the cutter slightly downward. Whether it's the straight, curved, or radius scraper, anglling the cutter downward slightly makes a big difference.
Getting the height of the cutter as close to the middle of the piece as possible also avoids catches. The slight angle downward is just some insurance that catches won't occur.
The problem I had when I first got the Stabilizer and tools was that, when I got down to the bottom end of the cut (at the center axis of the piece) there would be some catching (or something that approaches catching) when you go past the centerpoint of the diameter of the piece. One thing to help remedy this is to drill a 1/2" or larger hole to the depth that you want to hollow. Then, each time you get to that point, you can feel the hole and won't go past it.
I have taken Trent's in-person classes and that was a great help to develping the basics of hollowing and for using the Stabilizer. Since that's a little bit of an investment in course fees and travel, I'd advise viewing whatever Trent has on YouTube concerning hollowing. Also, I think Trent sells access to his online hollowing demos for a reasonable price. In all of his hollowing videos, regardless of the piece being turned, he goes through the hollowing process he follows. I have found that following those steps (with whatever variations you might find useful) is a good way to avoid any catches and to get the most from the Stabilizer. You don't need all the extras he uses in his videos, as there are always ways to do the same steps with what you already have access to.
I don't know if this helps at all. I'd say that hollowing and using the Stabilizer are really a learning process. The more you do it the better you get. The YouTube and other online videos are useful as starting points.