Contact Mover allows you to sync or move contacts between any pair of contact accounts on your iPhone.
To add a new account, simply go to your iPhone settings > Passwords & Accounts > Add Account. Once you add the new account, return to Contact Mover and you will be able to select the new account.
For those of you trying to sync your work contacts, as of iOS 11.3, unmanaged apps can no longer access contacts from your managed corporate accounts unless your IT department allows it. So if you have contacts in a corporate managed Exchange Server (Outlook) account, that account will not appear in Contact Mover unless you either:
a) Install Contact Mover as a managed app with the help of your IT department
b) Ask your IT department to "allow unmanaged apps to access managed contacts" as described here: https://support.apple.com/en-us/HT208749. For those organizations using MobileIron, see this article: https://www.mobileiron.com/en/blog/Why-iOS-12-matters-to-your-organization
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: