We extend the theory of Kazantzis-Kravaris/Luenberger (KKL) observers. These observers consist in immersing the system into a linear stable filter of the output with sufficiently large dimension and appropriate structure. After discussing the uniqueness of such an immersion, we provide two main results about its existence. The first one extends a known existence result by generalizing the structure of the target linear filter and reducing its dimension. While this approach relies on a generic choice of a sufficiently large number of distinct eigenvalues in the filter, we then propose a second existence result in the novel symmetric case where instead, the target filter is a cascade of a sufficiently large number of one-dimensional filters sharing the same eigenvalue. Finally, we propose a new cascaded procedure for the design of KKL observers. This method can be used in two ways: either to pre-filter a noisy output before using it in the observer, or to simplify the construction of the observer when the system can be written as the cascade of a nonlinear system and a linear one.