Airmail allows you to re-sync/update your contacts to Airmail if in some case, Airmail does not update itself (Airmail updates the contacts on each re-start). However, Airmail has an option for enabling you to update the contacts manually yourself. See the following screenshot with blue box below:
Once you’ve pressed the button Airmail will automatically fetch the new and updated contacts from your native Contacts.app or from the Gmail Contacts. This will also entirely rewrite the contacts database in Airmail.