In this case, I found her in the bottom of the cage horizontal (a large 36"x 24"x 24" butterfly cage with potted plants and supplements for them). I think she might have fallen, or maybe she just started out that way. I moved her out of the cage so she wouldn't be injured and put her in a safe place in a container on a seedling mat to keep her warm. I was concerned that she wouldn't moult properly, but she did quite well. I did have to help free one leg, but her wings and limbs did fine.
The moultings I normally see are vertical, as in this S. obscura in my backyard:

Interesting, I find that true crickets and grasshoppers can often molt fine horizontally and even inflate their wings properly that way, but katydids absolutely need to hang to molt, or they mismolt. The same seems to apply to two other similar groups, noctuid moths and butterflies. Butterflies need to hang, but the noctuids can eclose on the ground.

Yes, my Gryllidae (Gryllus texensis and Gryllus veintinueve) just suavely walk out of their exuviae laughing. They moult more easily than any other creature I've seen. My beetles and true bugs moult with considerably more difficulty.
My Scudderia furcata moult vertically, hanging from plant stems or sometimes on the side the net. I generally find only the legs of the exuviae: they might eat the rest?
The other katydids I've raised are Orchelimum silvaticum and Microcentrum rhombifolium (vertical moults). My only Conocephalus strictus was already a full adult (I brought her inside on 10.23.2020, and she's still alive and kicking - with one hind leg - on 04.12.2021).
My overwintering Neoconocephalus triops, two males, were brought inside 11.10.2020 as full adults, so I didn't see them moult. I love their sounds and wish I could find them some wives!
I'm hoping that some of the eggs laid in stems by my O. silvaticum will hatch. I check every day, but haven't seen any babies yet.

Wow! That conocephalus is really long-lived!

I've been able to observe the ageing process with all of the creatures I raise since they generally live long lives with me. The strictus lost one hind leg and got darker with age, but she is still moving around and eating. She's in an assisted-living cage where she can access her food on the floor of the cage.
Gryllidae start losing body parts: hind legs, segments of forelegs, cerci, antennal segments. I have a Gryllus veintinueve right now in a special hospice cage who is pretty much just an abdomen/thorax/head. He can still eat his lettuce!
My Pediodectes bruneri male moulted to full adult on 08.25.2020 and died on 02.25.2021, a full six months as an adult!
My Microcentrum rhombifolium moulted to full adult on 07.30.2020. His last song was 02.05.2021, and he died on 02.07.2021, so a bit over six months as a full adult.
In these two cases, I failed miserably in my desperate attempts to find them wives, and since mating seems to be so expensive for male katydids (and Oecanthinae, much more so), it's possible that the long lives were attributable to their never having mated.
One thing I learned last season is that while most of our katydids are univoltine, our Scudderia furcata are bivoltine here. I got some babies in September.
Unrelated, but one of my Deloyala guttata beetles named Orangie whom I isolated after some time to watch his life cycle moulted to full adult on 07.08.2020 and died on 02.12.2021 (over 7 months as full adult), and I've got more whose age I don't know but they've been with me since late summer / early fall. My very first bug of any kind was Angel, a full adult Oncopeltus fasciatus who came home with me on a milkweed plant on 10.22.2017 and died on 05.02.2018. I don't know how old he was when I adopted him, but he lived a nice long life and had two wives. :)

That means so much to me! I've got several more coming, several different creatures raised last season.