We say a set-valued u.s.c. function $f$ from $[0,1]$ to $[0,1]$ is a Smith function if $f$ is surjective, the graph of $f$ is connected, and the graph of $f$ is the union of finitely many horizontal and vertical line segments. The author introduced inverse limits with Smith functions in a presentation at the 2021 Spring Topology and Dynamical Systems Conference. Later, in a 2023 paper, the author answered some questions posed by audience members at that 2021 talk, and he raised some new questions as well. This presentation at the 2025 Summer Topology and Its Applications Conference will discuss our further progress on the study of inverse limits with Smith functions, including some new results and conjectures. Our focus will be the case where the inverse limit is a continuum, in which case we wish to determine when such an inverse limit could be indecomposable.