| Cheap Flights: Austin to/from New Orleans $138 r/t [May-Nov] â Southwest Southwest has round-trip flights from Austin (AUS) to New Orleans (MSY) for $138 with a connection. About $69 each-way. Flights May to November. Origin/Destination reversible. Carry-on and checked bag are allowed for free. Southwest doesnât charge change fees. Use the fare calendar links to pull up the fares. Click for more details and booking links Cheap Flights: Austin to/from New Orleans $138 r/t [May-Nov] â Southwest . | 