  Recent experiments have produced mounting evidence of Majorana zero modes in
nanowire-superconductor hybrids. Signatures of an expected topological phase
transition accompanying the onset of these modes nevertheless remain elusive.
We investigate a fundamental question concerning this issue: Do well-formed
Majorana modes necessarily entail a sharp phase transition in these setups?
Assuming reasonable parameters, we argue that finite-size effects can
dramatically smooth this putative transition into a crossover, even in systems
large enough to support well-localized Majorana modes. We propose overcoming
such finite-size effects by examining the behavior of low-lying excited states
through tunneling spectroscopy. In particular, the excited-state energies
exhibit characteristic field and density dependence, and scaling with system
size, that expose an approaching topological phase transition. We suggest
several experiments for extracting the predicted behavior. As a useful
byproduct, the protocols also allow one to measure the wire's spin-orbit
coupling directly in its superconducting environment.
