When you are looking to make a small book-in-a-book as part of a pop-up spread, there are a few techniques that can be applied. One is called a "Waterfall Mechanism". What makes this mechanism so much fun is that when a tab is pulled, it automatically flips each page in sequence. Below are a few links and videos explaining the concept. They are all by people who have shown their processes on the internet.
First is an interesting tutorial by Srushti Patil. She demonstrates the steps needed to make what she calls a "Filing Card". The concept can be applied to something that is smaller, and placed well within the page of your book, rather than being a stand-alone pop-up. The pull tab in her example is large, sticks out the side at all times, and is meant to make a visual statement.
6-minute video by Srushti Patil
Below is another demonstration making a set of pages that are all of equal size, and a lot of them. This video is very detailed, and created by Sam Calcott. She makes a strong and fast moving mechanism because of the narrow width of the folded strip. the narrower the folds, the faster the mechanism. So if you want to regulate ths speed of the mechanics, this is how you can do so. Also, because of the narrow folds, the pages are offset only a little bit.
20-minute video by Sam Calcott
Here is a tutorial video by, "Simply Cute Ideas", showing the concept of equal size pages, but with a bigger fold on the pull tab, thus more of an offset between pages. This video has only music playing and no speaking. It is also quick. Really pay attention to how the fold width affects the visual and speed aspects of the mechanism. Both this tutorial and the one by Sam Calcott show the pages as an offset stack, but if the first "cover" page was longer, it could overlap all of the subsequent hidden pages to give the effect of a closed book-within-a-book.
4-1/2-minute video by Simply Cute Ideas
Lastly, here is an example of how the offset stack can be incorporated into the visual design of the art. Note how each layer peeks out and hints at what's to come. Also, each layer is cut into a shape that corresponds with the art rather than being a precise rectangle. Only one edge actually needs to be straight, and the rest can be any kind of wonky shape you need. Pay attention to how the artist has incorporated the text into the mechanism by integrating it on the back side of each flap. This is a really clever way of conveying information in a clear, interesting manner. The words are concise, made to fit the size and shapes of the flaps, and in a few cases accompanied by little sketches. For more information on incorporating text into your book, see the Text As a Compositional Element and Text: Case Studies pages.
1 minute video by Paper Paul