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 > Accounts & Passwords > Add Account. Once you add the new account, return to Contact Mover and you will be able to select the new account.
As of iOS 11.3, Apple has made it so that unmanaged apps can no longer access contacts from your managed accounts. So if you have contacts in a managed Exchange Server account, that account will not appear in Contact Mover unfortunately.
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: