A very nice generalization of indexing by a well-ordered set and using induction to prove something is provided by the notion of a $\kappa$-good tree for $\kappa$ a (usually infinite) regular cardinal [see this MO question](https://mathoverflow.net/questions/28871/k-good-trees-and-k-compactness-of-colimits-over-k-small-downwards-closed-subposet).  To see it put to use in homotopy theory, take a look at Lurie's _Higher topos theory_ A.1.5, A.2.8, and A.3.3, where it is used to prove (indirectly) that functor categories (and simplicially enriched functor categories) A^C where A is a combinatorial (resp. combinatorial simplicial) model category admit fibrant and cofibrant replacement functors (this follows by the small object argument once we prove that A^C is itself combinatorial).  

The key here is the _regularity_ of the cardinal $\kappa$, which allows us to take downward closures of $\kappa$-small subsets with impunity as well as enlarge subsets until certain diagrams commute (that is, given some diagrams commuting in the limit, we can find $k$-small subset B for which the diagram commutes in the limit of B.