  We progress finite-size scaling in systems with free boundary conditions
above their upper critical dimension, where in the thermodynamic limit critical
scaling is described by mean-field theory. Recent works show that the
correlation length is not bound by the system's physical size, a belief that
long held sway. Instead, two scaling regimes can be observed - at the critical
and pseudo-critical temperatures. We demonstrate that both are manifest for
free boundaries. We use numerical simulations of the $d=5$ Ising model to
analyse the magnetization, susceptibility, magnetization Fourier modes and the
partition function zeros. While some of the response functions hide the dual
finite-size scaling, the precision enabled by the analysis of Lee-Yang zeros
allows this be brought to the fore. In particular, finite-size scaling of
leading zeros at the pseudo-critical point confirms recent predictions coming
from correlations exceeding the system size. This paper is dedicated to
Jaroslav Ilnytskyi on the occasion of his 60th birthday.
