The information below is an excerpt from a post on the St Barths forum, SBHOnline. Poster thomasj has done an excellent job of ferreting out flight information.

American Airlines
MIA - SXM: resumes Oct 4 (1 daily until Dec 15 when 2 daily resumes, note: Nov 6 flight arrives SXM one hour later)
JFK - SXM: resumes Dec 15 (1 daily)
PHL - SXM: resumes Dec 15 (1 daily)
CLT - SXM: resumes Dec 15 (2 flights daily, 3rd flight on Saturdays)

You can find the complete post, including the other airlines, at https://www.sbhonline.com/forums/showthread.php/91833 Be sure to go to the last post in the thread for the most recently posted information.