You can only be in one alliance at a time. To join a new one, you'll need to leave your current alliance first.
Steps to Change Your Alliance
Open the Alliance screen.
Go to the Settings menu.
Tap Leave Alliance.
Confirm your decision.
Wait for the cooldown period to end.
Find a new alliance and tap Join Alliance.
What Happens After You Leave
You won't be able to join another alliance for 24 hours. Your alliance progress will remain with your previous alliance. Your personal contribution will be reset.
Joining a New Alliance
If it's an open alliance, you'll join instantly If it's closed, your request must be approved by the leader.
Notes and Tips
Make sure you really want to leave before confirming, as the cooldown starts immediately. Plan ahead so you don't miss out on alliance rewards or events.