From a (very quick) look at Expert Flyer it would appear that on every day of the week at the end of March there is an SA fight around 5pm which would connect with your JFK flight. Which day are you looking for?
I don’t think you’ll get any better result than with the US call centre which by all accounts is quite good (I hive no personal experience with them).
Is this a paid ticket or an award. If the latter it would complicate things (I have a similar problem with a CPT - JNB flight, but fortunately without a connection)