According to this post in God Save The Points and this post in Frequent Miler, American Airlines and British Airways are in a fare war with Delta (and maybe United?) for trans-Atlantic PREMIUM ECONOMY passengers.
Premium Economy is much more comfortable than regular Economy for long flights, especially overnight flights like most from the US to Europe. Premium Economy specifics vary between airlines (and type of aircraft), but the seats typically have more legroom, and may be wider, recline farther and provide better food and beverage selections.
To find these deals, search Google Flights from Austin and also from cities that you can reach on Southwest from Austin, like Boston, Miami and New York. Be sure and use this important tip to force Google Flights to show you the actual best prices.
You are likely to find these best prices for travel in April, May, and August through December.