I don't remember signing up for the LA specials, so I think they signed me up when I bought an online ticket with them. Anyway, they are offering a JFK-SCL special this week for $399 and a MIA-LIM special next week for $299. The Miami isn't the greatest for a mileage run, but $400 to Santiago from New York is pretty decent.
Also, there is a different special, but I don't know exactly what is valid. The wording is "Buy your tickets on LanChile.com throughout August to travel from the United States any time in the next year and you will receive a 5% discount." It looks like you need to start travel in the US, but given that LA's translations can occasionally be suspect, it might intend to mean that you need to buy the ticket from the US. Don't have time at the moment to look into it more.