Based on my research, there seems to be no way to increase the actual odds of getting a shiny Pokémon in Pokémon Sleep. Game Rant makes it seem that maximizing Snorlax’s Drowsy Power increases Shiny odds, but all the other sources don’t seem to agree with that, so I’m guessing Game Rant means increasing the amount of Pokémon encountered which increases the chances of finding a shiny Pokémon because the quantity of Pokémon has increased.
What you do to increase your chances of getting a shiny Pokémon is just simply increase the amount of Pokémon you encounter during your sleep by raising Snorlax’s sleeping power. You do that by feeding Snorlax through the day and……… well sleep. It seems though the Eevee event coming up gives Eevees with an improved Shiny rate. That’s about it unless the secret depths of Reddit that I cannot access for some reason has discovered something.
Game-Rant
Charlie-Intel
TheGamer
Game8
