| Nonstop Flights: Austin to New Orleans $128-$138 r/t [February-May] â Southwest Southwest has round-trip flights from Austin (AUS) to New Orleans (MSY) for $128-$138, NONSTOP. $64-$69 each-way. Flights between February and May. Origin/Destination reversible. Click for more details and booking links Nonstop Flights: Austin to New Orleans $128-$138 r/t [February-May] â Southwest . | 