New York seems like a tough place to start a TWA mileage run if the end-goal is three cheap segments.
Pick some city beyond StL that you actually want to spend a weekend in, and buy a ticket from TWA's website. It'll cost ya a couple hundred dollars or so.
I would have it much easier: I'm in Kansas City and have a few friends in St. Louis I could visit. If I had never flown TWA before, I could get this one taken care of with two $99 RTs (*very* restrictive coach tickets). Heck...I'd do it four times and get the 25,000 bonus miles, too.
As much as I sometimes hate flying this airline, they ALWAYS have the best promos!