  We show that the gradient of the $m$-power of a solution to a singular
parabolic equation of porous medium-type (also known as fast diffusion
equation), satisfies a reverse H\"older inequality in suitable intrinsic
cylinders. Relying on an intrinsic Calder\'on-Zygmund covering argument, we are
able to prove the local higher integrability of such a gradient for
$m\in\left(\frac{(n-2)_+}{n+2},1\right)$. Our estimates are satisfied for a
general class of growth assumptions on the non linearity. In this way, we
extend the theory for $m\geq 1$ (see [GS16] in the list of references) to the
singular case. In particular, an intrinsic metric that depends on the solution
itself is introduced for the singular regime.
