It was definitely alive when I found it. I followed instructions closely for cradling it, but, to my disappointment, it never hatched.

they are designed to be almost vertical, at about a 15 degree angle. The pointed end is attached to a vertical surface, the head end is suspended by a silk girdle and does not touch the surface. Suspended in a such a way the dorsal surface of the pupa is more ventral than the ventral surface of the pupa. Thus, when the butterfly emerges its legs can grab onto the vertical surface and the butterfly can crawl upward to expand and dry its wings. So, make a small paper tube, about one-half the length of the pupa, and angle this (at about 15 degrees) to a vertical rough surface making sure that there is sufficent space between the ventral head end, where the legs are, so that the butterfly can stretch its legs when it emerges. Place the pupa in this sleeve, head up and legs towards the vertical surface. This system has worked well for me with Canadian Tiger Swallowtails.
This is a situation where a simple drawing would do better than my wordy explanation.

I tried this with a Black swallowtail pupa this year and it's wings, as well as part of the body, were badly deformed when it emerged. I was careful to follow directions closely, as well as looking at some images of how to do it. I was careful not to make the tube too tight. I don't know what I might have done wrong. Any ideas? Next time I come across a pupa needing help, I'll try the glue gun and see how that works for me.

it was bruised sometime, and may have had nothing to do with how it was hung or placed. The hot glue method works for the ones that hand "right side up" with a girdle too. Many butterfly houses hang the pupae upside down from the cremaster regardless of species, and they all seem to hatch out just fine (basically it's the same as if the girdle broke or slipped off in nature - which happens sometimes). The main issue is that if the glue is too hot it can damage or kill the specimen, and I don't know what is a threashold maxinum temperature. Some places use other adhesives that are not heated, but I don't remember what works well and what doesn't. It seems that several glues might work well, but then the issues include waiting for the glue to dry, and whether it will actually stick to a smooth pupa that might change shape or move. Also, if too sticky still when the adult emerges, it could get tangled up with the glue. There could be an issue of chemical toxicity with some adhesives as well. Silicon caulk works fairly well, but you have to let it dry before the insect can actually hang from it. There are some versions that are pastier than others right off, and avoid any that have anti-fungal agents or have something mixed in other than just the silicon caulking itself.

Another method that I have used, that works well (usually) for most species is to gently make sure all the silk and debris is cleaned off of the cremaster. Then the hooks on the cremaster will engage with fine fibers in loose woven materials such as paper towels, napkins, toilet paper, and some types of cloth. So, they can easily be hung from these materials as if it were their own silk pads. Sometimes the pupae help out by wiggling while you try to get them hooked on, which helps to tangle the hooks in the fibers better. If the hooks don't engage properly though, the pupae will fall.

Sometimes when you find a butterfly pupa in the wild and wish to bring it in, you can just break off a piece of what it is attached to, pupa and all. Or you can carefully peel the silk pad and girdle from the substrate and tape or glue that back on to a new support indoors. This is easier than it sounds, and leaves the insect hanging just as it was originally.

it sounds like a great method but I'm having trouble visualizing it. Does it adapt well for other species?

I should probably add the hot-glue method is what was used for all butterfly pupae at the butterfly exhibit where I used to work. They were actually glued to paper in large batches, and then the paper was pinned to the underside of a shelf. Nothing natural about it, but it worked very well. I like the idea of not having to mess with glue, though.

The hanging upside-down-pupae can be attached by the 'tail' using the hot glue method. I think most of the nymphalids pupate this way
, the Monarch and I believe some of the South American Swallowtails. It is used extensively in the UK butterfly houses where the pupae are stuck onto long 1/4 diameter wooden rods.
Some of the Pierids pupate head up and at an agle supported by a silk girdle, just like that of the Tiger swallowtail, and these would best be placed in a cone.

I often raise those, and although it's rare that I have a chrysalis not firmly and appropriately attached to the twigs I provide for them, occasionally they pupate on the cage and I have to move them. Next time that happens, I'll definitely give your method a try.

I used the method you described to suspend the swallowtail in a small, netted butterfly enclosure. I'll overwinter it and hopefully come spring a healthy Tiger (my first) will emerge. I'll be glad to send a photo of it in its "holder" if anyone says they want to see it. First, I'll have to figure out how to add a photo to a comment, as I haven't tried that before. Thanks a lot for all your help, it's very much appreciated.

I hope you get an adult to confirm the ID.