Once you have your game connected to your Game Center account, you can use this account to load your saved progress on another Apple device.
First verify that you original game is attached to your own Game Center account on your primary device.
Open your game and go to Settings in the upper left corner of your map. If your Game Center status is “Game Saved”, your game is attached to this particular Game Center account.
On your secondary device, go to your device Settings > Game Center and sign in to the same account. Install and open the game, and it will prompt you to load your original progress automatically after the opening.
If not, go to the settings in the upper left corner and tap “Connect” to make the popup appear.