Indeed physically impossible to get a later flight and home in the same day with AA. Looks like you can:
1) Stick to what they put you on (an early morning yes but you only really miss out on the morning).
2) Overnight via MIA, JFK/LGA or BOS. I might choose the JFK->LGA route (December in NYC is great ... but hotels are pricey).
3) Ask for a credit/refund and if you have miles on DL or US ... there are afternoon departures on both carriers -5:10pm and 4:00pm respectively.
3a) I doubt they would do this but you can try to see if they will route you on AA (STT-JFK) and DL (JFK-STL). It's the 1pm departure out of STT so not too much of a real option ... but never hurts to ask.