I have an Iphone 6 and I dont have the Start New Conversations from option I do not have the start new conversation from option, I was close to fixing this issue following your direction, and then I hit this issue. Its just not there it ends with the "add another email..." option. This is so aggravating its confusing all of my messages to everybody, its caused me so much verbal pain and misconstruction. LOL...i think im laughing out loud i dont know
If you are showing a conversation, go back to Messages.  (Top Left on your screen)  Then at the top right side you will see an icon to start a new message.

See picture below.

Apple Support has more help for you too.
Hope this helps you.