These seem like two (or more) different things.
Things that take outside tools to solve: E.g., an “extracurricular research” puzzle where the player is expected to look up information outside the game. I actually really like this sort of puzzle, but the danger is that the player will not realize that they are meant to do this sort of research and get frustrated that they cannot find the relevant information in-game. So, this takes the correct context or careful hinting.
Bad: “I have to look up a book by name, but I don’t know the title and I can’t find anything in-game that tells me. Does this game just expect me to know the complete works of Rainer Maria Rilke?! (consults walkthrough) How was I supposed to know to Google that?! This game sucks!”
Better: “The game mentioned early on that it might be helpful for me (the player) to have some knowledge of the mating habits of the golden bowerbird. I didn’t, so I looked it up, and now I have both solved a puzzle and learned something new.”
Even better: “There is a message in Morse code. I don’t know Morse code, but that’s clearly what it is, and I bet I could decode it by looking up a reference guide.”
Context-appropriate: The Barber of Sadville, a game created for an Easter Egg / ARG associated with the BBC sitcom The IT Crowd, at one point requires the player to know something about the memory map of the BBC Micro computer. However, if the player has even gotten as far as figuring out how to extract the game from the DVD’s subtitle files, they’ll probably assume that Googling stuff is fair game.
Knowledge you can only have after playing through a game once before. The “bad” version of this is what @Mike_G mentions, and Graham Nelson specifically highlighted in his “Player’s Bill of Rights”:
But this only annoys because it seems unfair. Conversely, in Aisle, the entire game is played in one turn, and repeating the game with knowledge of past events is the whole mechanic. You also mentioned The Wand, where there’s a whole second game that is only hinted at after winning. Both of these examples work because the extra content that is available on restarting feels like a reward rather than a punishment. Rather than communicating, “You did it wrong, sucker,” the game communicates, “You liked that? Now go back and you’ll find some more!”
It’s a matter of preference how hidden to make this meta-content. I like how The Wand does it: there is hinting, but it’s obtuse enough that most people who discovered the hidden content probably found out about it from reviews. Conversely, I’m not sure Ryan Veeder ever expected people to figure out all of the hidden meanings in Ryan Veeder’s Authentic Fly Fishing.