Booking Southwest flights on a Tuesday often proves to be cost-effective, as airlines typically announce fare sales early in the week. This pattern means that by Tuesday, you might find some of the best deals available. 

If you’re flexible with your travel dates, you can further increase your savings by adjusting your itinerary to off-peak days, like Tuesdays and Wednesdays. It's also beneficial to sign up for fare alerts from Southwest, which can notify you about price drops and promotions. 

Additionally, while booking, always compare prices across different dates and times to ensure you're getting the best deal. Remember that last-minute offers might also surface on Tuesdays, so keep an eye on those too!