  The intensity of the turbulence in tokamaks and stellarators depends on its
ability to excite and sustain zonal flows. Insight into this physics may be
gained by studying the ''residual'', i.e. the late-time linear response of the
system to an initial perturbation. We investigate this zonal-flow residual in
the limit of a small magnetic mirror ratio, where we find that the typical
quadratic approximation to RH (Rosenbluth & Hinton, 1998) breaks down. Barely
passing particles are in this limit central in determining the resulting level
of the residual, which we estimate analytically. The role played by the
population with large orbit width provides valuable physical insight into the
response of the residual beyond this limit. Applying this result to tokamak,
quasi-symmetric and quasi-isodynamic equilibria, using a near-axis
approximation, we identify the effect to be more relevant (although small) in
the core of quasi-axisymmetric fields, where the residual is smallest. The
analysis in the paper also clarifies the relationship between the residual and
the geodesic acoustic mode, whose typical theoretical set-ups are similar.
