  Zeeman-Doppler imaging is a spectropolarimetric technique that is used to map
the large-scale surface magnetic fields of stars. These maps in turn are used
to study the structure of the stars' coronae and winds. This method, however,
misses any small-scale magnetic flux whose polarisation signatures cancel out.
Measurements of Zeeman broadening show that a large percentage of the surface
magnetic flux may be neglected in this way. In this paper we assess the impact
of this 'missing flux' on the predicted coronal structure and the possible
rates of spin down due to the stellar wind. To do this we create a model for
the small-scale field and add this to the Zeeman-Doppler maps of the magnetic
fields of a sample of 12 M dwarfs. We extrapolate this combined field and
determine the structure of a hydrostatic, isothermal corona. The addition of
small-scale surface field produces a carpet of low-lying magnetic loops that
covers most of the surface, including the stellar equivalent of solar 'coronal
holes' where the large-scale field is opened up by the stellar wind and hence
would be X-ray dark. We show that the trend of the X-ray emission measure with
rotation rate (the so-called 'activity-rotation relation') is unaffected by the
addition of small-scale field, when scaled with respect to the large-scale
field of each star. The addition of small-scale field increases the surface
flux; however, the large-scale open flux that governs the loss of mass and
angular momentum in the wind remains unaffected. We conclude that spin-down
times and mass loss rates calculated from surface magnetograms are unlikely to
be significantly influenced by the neglect of small-scale field.
