Let's say that I start with a Stanford Torus that can house some amount of people. Over time, more and more people start living there, to the point it starts getting crowded. How could I expand the habitat without just building a new one?
I'm going to add a refinement to the previous answers which state you should stack extra rings in order to create more living space.
Different configurations of space colonies
As you can see, the preferred configuration for expanding the "Stanford Torus" is the "Banded Torus" at the bottom of the middle row. What yo should consider is even longer term.
If you continue to "stack" elements in a banded torus, you are creating a cylindrical structure. By placing end caps on the cylinder, and arranging a lighting system for the inside (for example a sun tube running down the spin axis) you have created a massive cylindrical habitat.
The inhabitants can continue to live in the banded torus, since it is already there, and provides radiation protection and a multiply redundant life support system. The space in the middle can be used for a multitude of things, including simply having it as a wildlife park to house threatened ecosystems, or for farming, or to provide even more living space.
This can even continue, end caps can be placed to create cylindrical living volumes, then more bands can be added before another cap is placed, providing a means to have multiple volumes in case of an accident or deliberate puncture of the structure, or to isolate sections to prevent the spread of disease or unbalanced ecosystem disaster.
You'd have to -- I think -- build another torus on "top" of the existing station (because ships probably dock on the outer edge, so you can't build there)
Importantly, the station would have to be designed for expansion, and that expansion construction would have to proceed in a very symmetrical manner (whenever you add a chunk of mass to one part of the station, you've got to add a similar amount of mass to the opposite side) so that you don't unbalance the station (thus possibly destroying it).
Build a second layer on it, do this every time you need to add volume until you have a large cylender. Ideally you could focus on designing each "layer" to be mechanically and functionally capable of total self reliance. Once you have a cylender too big to be feasible for more add-ons you split the layers as thier own independant stations to each begin the cycle anew. In this way your population stations can functionally "reproduce."