Many classification theorems (e.g. of the finite subgroups of $SO(3)$, or the finite-dimensional complex simple Lie algebras, or the finite simple groups) have some infinite lists, plus some "sporadic" or "exceptional" examples.
It was opined in Why/when classification of simple objects is "simple" ? E.g. (unknown) classification of simple Lie algebras in char =2,3… that any such theorem only reflects our current knowledge of the subject. One might hope that better understanding would lead to an alternate way of slicing up the set of examples, with fewer sporadic cases.
Are there actually examples of this happening? I want a case where the initial classification is actually complete and correct; it's only our human description that has improved.
Two non-examples: (1) As I understand it, Suzuki found an infinite family of finite simple groups, that we now regard as twisted Chevalley groups for $G_2$ and its outer automorphism in characteristic 3. But that was before the classification was complete. (2) Killing had two root systems that turn out to both be $F_4$. So he wasn't quite correct.
Best Answer
"Classification" might be a very strong word for this example, but I think real quadratic polynomials underwent something like the development you describe. They are classified by their discriminant: positive for two distinct roots, zero for a double root and negative for no roots ("three families"). The generalization of complex polynomials and roots simplifies this: discriminant zero for a double root and non-zero for two distinct roots ("two families"). When you specialize this to real polynomials you end up with what you started with, with the added bonus of two complex conjugate roots instead of none in the case of a negative discriminant.
Surely this is only a toy example, but it does show that broadening the viewpoint can reduce the number of cases to consider.