iOS 10 Music App: "Add to up Next"?
There appears to be a bug in the way the iOS 10 Music app handles this. However, there is a workaround that will get you by until Apple fixes it.

Take these steps:


Go into the Up Next screen
Turn off the Shuffle function
Completely close down the Music App (don't just exit)
Launch the Music App again
Use 3D Touch (press firmly) on a song and select the Play Next option
Now use 3D Touch again, but on another song, and you'll see the Play Later option. 
Select Play Later to add that song to the bottom of your Up Next queue.
Now the Play Later option will be there for all following songs.


However, the reason this is a workaround is that:


you no longer have the Shuffle function available to you.
I'm not sure what the workaround is if you have a device without 3D Touch


To have this whole issue fixed, send feedback to Apple.