What model iPhone do you have?
If you have an model with force touch (6S or higher) search your contact in spotlight, long press on the resulting name, tap on the contact name again it will open in the contacts app where you can edit it.
If you don't have an iPhone with force touch then spot light contacts.app, and search the persons name from there. Yeah, not as easy but it might be easier then searching through all your apps looking for Contacts.
Amazing. I never knew this. Though you should still be able to edit or at least open up the contact in the full translucent spotlight thingy. Are they called cards? I don't know. Anyway, you should be able to open up the editable contact from there. But this tip is super helpful. Thanks!