I have noticed timing is everything! Until now I have been a part of a studio and so I had different habits that lent itself to better timing- for example, I set a piece aside after throwing it and let it dry for a bit without being wrapped. In my own studio I can come and go with more frequency so I end up wrapping my pieces earlier. I do wire twice, once right after I throw and once when I am trying to lift the piece from the bat.
I have been using cups, bottle, etc to try to restore the shape. I had no idea this tapering tool existed! Will have to try that too.