As per the documentation of Facebook messenger platform here, the welcome screen appears only once for a user when he/she starts the conversation. It will not appear for the same user ever again on mobile.
From the docs:
The Welcome Screen only appears for new conversations. On mobile, deleting conversations will not cause the Welcome Screen to appear again. On desktop web, if you delete the conversation, you can make the Welcome Screen appear again.
So, if you want to see the welcome screen again I'd suggest to add another user as a tester in your Facebook application (https://developers.facebook.com/apps/YOUR_APP_ID/roles/) and start a new conversation but don't click on the "Get Started" button unless you're done editing this screen because it won't appear for this user again.
Edit: 2016-07-26
Facebook has removed the documentation pages previously mentioned in this post. Now, there's a new page on Get Started
button here, which says:
The Get Started button is only rendered the first time the user interacts with a Page on Messenger.