You can set your contacts to sync between your iPhone and Mac using a service such as iCloud or your primary email account.
First you want to make sure Contact syncing is enabled for iCloud or the account you choose. On the iPhone, go to Settings > Mail, Contacts, Calendars. Select the account you wish to use for syncing and ensure the Contacts option appears and is turned on. Do the same on the Mac in System Preference > Internet Accounts, ensure the Contacts option appears and is checked. Then you want to set the default account for new contacts. On the iPhone go to Settings > Mail, Contacts, Calendars > Contact Section > Default Account (If Default account does not appear between the "My Info" & "Contacts Found in Mail" then you only have one syncing account enabled and it is automatically the default, so disregard this step on the iPhone.) On the Mac, open the Contacts App, then from the Menu bar, go to Contacts > Preferences > General Tab > Default Account (If Default account does not appear below "Address Format" then you only have one syncing account enabled and it is automatically the default, so disregard this step on the Mac.) Your newly added contacts should now sync between your Mac & iPhone (both ways).