It doesn't. "Despidió" is already past tense. That said, "despidiose", while valid, is quite archaic. If it was me, I'd say "se despidió demasiado" or even "se despidió de más"
Edit: Reading the meaning of the song, I'd say "dijo adios demasiadas veces", as it stays closer to the original meaning.
Edit: Reading the meaning of the song, I'd say "dijo adios demasiadas veces", as it stays closer to the original meaning.