I start looking at Bookingbuddy.com for flight pricing three months before my trip. Last year, in September 2016, the cost 90 days ahead of the flight JFK to Santiago was around $600. I waited until October and ended paying approximately $900 round trip with a stop in Miami on American Airlines, coach, from JFK in New York. This year, 2017, I started 90 days out, and pricing was $1300 on American, but my budget was only $900. I found that Avianca had seats in my price range and booked with them in September. I have never flown with Avianca. American Airlines is what I call a get on, get off the flight, no problems and no issues, clean, friendly, professional with terminals that are organized and calm, and if you are lucky, some good movies. I will report back later on my experience with Avianca.
I discovered that Avianca makes one or two stops down to Santiago from JFK, one in Bogota Columbia and the other in Lima, Peru. If I wanted, I could get off the plane at each stop, spend days there, then get back on and continue to Chile. I declined to do this as I have too much luggage, but it is a consideration. If you do fly with them, it might be interesting to check this out.
Each Airline on international flights allows you to check 2 pieces of luggage that weigh less than 50 lbs and one carry on. Please check before you assume I am right or the rules are still the same! The luggage should have Big wheels and locks.
Here are a few links.
Update December 2020; check out Copa airlines, it is a great airline with flights to Santiago through Panama City.