LTTRAVEL2: Welcome to FT. And Welcome to AA.
Your tickets should be fine, since Hartford and JFK are not co-terminals. I'm not positive you'd have a problem if they were, but I always avoid returning to a co-terminal when I do what you're planning to do. The list of co-terminals can be found in the fare rules on Travelocity; here is a copy:
CO-TERMINALS -THE FOLLOWING GROUPS OF CITIES ARE CONSIDERED TO BE THE SAME POINT BUR - SNA - LAX - ONT - LGB, EWR - HPN - ISP - NYC, OAK - SFO - SJC, BWI - WAS, FLL - MIA, HRL - MFE.
As you can see, it might be risky to return to Islip, Westchester or Newark (and of course, as you know, forbidden to return to JFK) but Hartford should be ok. Have a great run.
As an aside, whenever I do this I always end up imposing on a friend to retrieve me from the alternate airport (or drop me at the originating airport or both). Those friends or relatives are called "Enablers."
Shame to see another promising life wasted to addiction.