I normally raise my 3rd-5th instar cats in 24 oz containers with special Poly Fabric lined lids that allow air flow, but still keep humidity up much higher than in an open air cage. Plus, they are easy to bleach/disinfect/clean so they can be safely reused. I never have to mist. Since I switched to them a few years ago and stopped misting, my cat death rates dropped by a lot.
The sturdy fabric on the tops of the lids usually provide a very secure base for the silk to hold on to. Occasionally, a cat decides to rush the chrysalis process and falls to the bottom of the container.
To rehang the chrysalis: I take cotton string and cut them into 8" long pieces. I then separate the strands of the string and tie an individual strand around the base of the cremaster. After that it's just a matter of tying the string to something to hang the chrysalis. For the containers, I tape the ends of the string to the outside of the containers and then put the lid on to hold the chrysalis in place.
At work I usually set up a temporary public display with reptile window screen cages. It's easy enough to have branches in those cages that the chrysalis' can then be tied to if needed. If I run out of space in my rearing containers, I sometimes just throw a bunch of 5th instars into the screen cages and they will make their chrysalis' on the top or on the branches. This year though I don't have a public display because the offices are being renovated and the dust/mold stirred up plus the paint fumes are nasty. Don't want to subject the babies to that so am raising them all at home in the cups.
Some pics from last year: