Contact Mover allows you to perform a sync or move operation on any pair of existing contact accounts on your iPhone.
To add a new account, simply go to your iPhone settings > Contacts > Accounts > Add Account. Once you add the new account, return to Contact Mover and you will be able to select the new account.
For more help on setting up a new account, go to https://support.apple.com/en-us/HT201320
Video showing how to add a new account: