If you took the LCD screen from a tablet. Segmented it into equally sized rectangles, and then added connection ports on the sides of each rectangle. Could you then separate and stack the rectangles on top of eachother to save space? And then fold it back out to create a tablet-sized screen again.
(Crude drawing of what I mean)
There is a foldable LED screen at CES this year that works similar to how you describe but 4 panels horizontally. The four panels fit so well that no join is visible.
https://youtu.be/A5bHCXRxAMg?si=AYEJ_HC5nFrjQHps