Currently, it only shows at the very start of the game, when you complete a goal and since beta2 when you rejoin the game.

It should additionally show up when an opponent team completes a goal, and when you die.

Possibly it would also make sense to send the message on a timer periodically in order to further minimize players losing access to it or getting confused on how to return to the lobby.

One idea is perhaps to prefix the tellraw messages in chat with an exit symbol that has a hover/click event to return to the lobby.
Something like an arrow pointing to a door maybe?

e.g.

Pros:

• shows up more often without adding additional spam to the chat.

Cons:

• might not be often enough if there are a lot of users typing in chat?
• might not be immediately obvious to the user, unless it is explained to them beforehand.

Item gets are quite frequent, so if many people are chatting, you should also expect more item gets, don't think that would be an issue.

I just think it's not obvious enough and it's not really in a place where you'd expect it.
It becomes even more convoluted when more actions get added such as entering spectator mode (#70).

I like that idea.
I think I would still prefer to send the full message in some situations (Rejoining I don't see any other way since there is no message send then anyways), but I also think I'd prefer it after completing a goal since it's very common to use these options then.

The 3 bars should be added to the GO! message on game start and every item get message. If you have other places in mind as well, there might be other things I am not thinking about.
Also, not sure when the Hint: text should appear now. Currently it is intentionally only on game start to reduce chat clutter, but if it's player triggered, I guess it could just always be when you press the 3 bars (but not when the full message is send by default)

Beside deciding where to show current hint-lobby,
it'd be also good to add hint-menu (about using ≡) that displays for sure for the rejoining players, game start and maybe in other moments

How about:

1. Remaking the message a little bit, so it is one line,
   that is sent instead of "Click here (...) lobby. Click here (...) spectator", like:

   [Bingo Menu ≡]

2. Additionally make an icon - idea with prefix above (both chars already in MC fonts):

   • [☰] (U+2630, Trigram for Heaven) - in /nonlatin_european.png
   • [≡] (U+2261, Identical To ) - in /ascii.png

   Same order as above:
   
   I think most people know three bars already ;)

And after clicking those - more options are sent to player (back to lobby, spectator, etc.), not making the chat cluttered when you don't use it.
+It allows to easily add more options that you can change on the fly during gameplay.