Tip number 1 Don't panic! Take a breath and assess the situation.
Tip number 2 Always use the airlines's app when flying. If you don't want to keep it long term just delete it afterwords. Especially for the domestic airlines many of these changes can happen automatically right in the app. Delta and United are have particularly good app in case if irregular operations (irops). Often your problem can be solved with one click of a button to accept the proposed changes.
Tip number 2. If the app doesn't solve the problem get in line for the gate agent or at the customer service desk. But, while you're waiting in line call the airline's 800 number. In case of irops the Customer service agents can rebook just like the agents in front of you. You're essentially standing in two lines at once. Whether reach a phone rep or the desk agent they will do all they can to get you rebooked quickly.
Tip number 3 if the options they are giving you don't seem very reasonable, ask if they can book you on a different airline. Most airlines can but they wont' always do it without you asking. (I suggest looking at flight options on google flights to know your options before you reach a human).
Finally, tip number 4. Be extra nice to the agents you speak to. They didn't cause the problem and have probably been getting yelled at by every passenger. It's their job to help you, but if you're nice they will work harder for you! The Golden Rule definitely applies here!
In summary keep calm, hedge your bets and be prepared with options.
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