Southwest Airlines is offering 30% off many flights this summer and fall, but the deal isn't available on every ticket.

The past few years have not been kind to Southwest Airlines from a public relations' perspective. Thirty percent off your flight could help restore faith.

The airline said that its cardmembers will get their first checked bag for free, for up to eight other passengers traveling under the same booking. Southwest previously said that it would still offer two free checked bags to its Rapid Rewards A-List Preferred Members, and to customers who bought Business Select tickets.