  We consider the number of critical points of a stationary planar Gaussian
field, restricted to a large domain, whose heights lie in a certain interval.
Asymptotics for the mean of this quantity are simple to establish via the
Kac-Rice formula, and recently Estrade and Fournier proved a second moment
bound that is optimal in the case that the height interval does not depend on
the size of the domain. We establish an improved bound in the more delicate
case of height windows that are shrinking with the size of the domain.
