Let’s not forget that in 1937, Bilbo also said “good-bye” to Gollum!

• 04.003 although when they had said good-bye to Elrond
• 07.094 and said good-bye to Elrond.
• 07.145 Then at last they said good-bye to their ponies
• 07.147 Good-bye!’
• 07.147 And good-bye to you all,
• 07.147 good-bye!
• 07.152 Good-bye!
• 07.153 Good-bye then,
• 07.153 and really good-bye!’
• 07.153 Good-bye!
• 07.154 O good-bye
• 18.038 for him to say good-bye to his friends.
• 18.039 Good-bye

Attested in OED, and whether it means “God be with ye” or “God buy (redeem) ye” is unclear.

“goodbye, n., int., and adj.” OED Online, Oxford University Press, June 2017, http://www.oed.com/view/Entry/79929. Accessed 13 September 2017.