Much as the title, really.

I’m trying to describe a separate space attached to a much larger one (think something like a private dining room connected to a larger restaurant space, or a games/pool room that’s attached to but separate from the larger pub floor area). Is there actually a specific word for this?

It’s not so much an anteroom, as this is supposed to be a room connecting the outside to a larger room, but is there a term for a room like an anteroom that doesn’t have a door leading outside?

Many thanks.