  Hadronisation corrections are crucial in extractions of the strong coupling
constant ($\alpha_s$) from event-shape distributions at lepton colliders.
Although their dynamics cannot be understood rigorously using perturbative
methods, their dominant effect on physical observables can be estimated in
singular configurations sensitive to the emission of soft radiation. The
differential distributions of some event-shape variables, notably the $C$
parameter, feature two such singular points. We analytically compute the
leading non-perturbative correction in the symmetric three-jet limit for the
$C$ parameter, and find that it differs by more than a factor of two from the
known result in the two-jet limit. We estimate the impact of this result on
strong coupling extractions, considering a range of functions to interpolate
the hadronisation correction in the region between the 2 and 3-jet limits.
Fitting data from ALEPH and JADE, we find that most interpolation choices
increase the extracted $\alpha_s$, with effects of up to $4\%$ relative to
standard fits. This brings a new perspective on the long-standing discrepancy
between certain event-shape $\alpha_s$ fits and the world average.
