When you are replying to a morning greeting, do you reply with "Good morning, too" or, "Good morning to you, too" or just, "Good morning"?
"Good morning" is fine (and standard); "Good morning to you, too" is also fine. I have never heard anyone say "Good morning, too".
It would probably go far over the top, and confuse people, unless they had either very good senses of humor or were fans of a certain famous fantasy author, to answer as Gandalf did to Bilbo Baggins: "What do you mean? Do you wish me a good morning, or mean that it is a good morning whether I want it or not; or that you feel good this morning; or that it is a morning to be good on?"
(Click the second link to see Bilbo's reply.)