How to see contacts that are suggested when composing emails?
The easiest way to add a contact from the iOS mail app is to see their address in an e-mail. Received or Sent does not matter.

Simply tap on their e-mail address, or the name it displays for them and a callout window will appear with the choices 'Create New Contact' or 'Add to Existing Contact'. Tap on one of those and add the details as necessary.

Note that if you already have them in your address bar, following this process will show all the contact information you have for the individual.