How to tap into a light switch whose junction box is already full?
What is the size of the box holding this switch? Is it 1-gang, 2-gang, 3-gang, etc?

Would a deeper box allow the new cable and connections? There are 3-7/8" deep 1-gang boxes with a 23 cu in capacity which is higher capacity than standard.

Alternatively you could replace a 1-gang box with a 2-gang old work box. Of course, this might be aesthetically unpleasing.