I recently got a very good looking caladium and it has many new leaves coming in. One of its leaves had died and I pulled it off as I was pretty sure it's because it's growing in so many new ones. But now one of its new leaves, that hasn't even fully uncurled yet, is not looking too good. It's pretty yellow, and it feels very thin and floppy. I'm not sure what could be causing this

Caladiums like heat and a lot of water, but not dense soggy soil. It looks like it may not be very close to a window. How does the spot and condition compare to those things?

Few people have much success long-term with Caladiums as houseplants. If I'm remembering right @skylark is one of the few. Maybe she will have something more to add the next time she stops by this site.

The pot is way too big for this: it is most likely in peat, so will stay wet longer and roots will rot.
you need to put the pot on it's side and very gently try to shake the root ball out to see the state of the roots. If you do not see lots of white roots at the sides and bottom, remove the extra soil that does not have dense roots and put the plant in a smaller pot. Hard to judge by the picture, but it seems the current pot is larger then 6".
do not try to remove soil between the roots: the less you mess with them the better. Granted, some roots will get damaged, but try not to play with them at this point.
There are different Caladium varieties, from very large to quite small.
When soil stays too wet with few roots it might even rot the tubers.
try to keep it very warm, preferably in western window close to the window where it can get long warm end-of-day sun. that would be best for indoors: they definitely need 3-4 hours of direct sun to grow well. Temperature matters a lot: 75-80F is best.

Just wanted to add that yr plant is just starting to leaf out: i see 3 stems with 1 leaf each. Most likely it is 3 corms with 1 leaf ach; that is just starting to grow. normally they should have 3-5 leaves each.
So the rule is like with waking up bulbs: water very sparingly until you have a couple of leaves (and in a small pot at that).
My guess you overwatered it.