  A natural notion of higher order rectifiability is introduced for subsets of
Heisenberg groups $\mathbb{H}^n$ in terms of covering a set almost everywhere
by a countable union of $(\mathbf{C}_H^{1,\alpha},\mathbb{H})$-regular
surfaces, for some $0 < \alpha \leq 1$. We prove that a sufficient condition
for $C^{1,\alpha}$-rectifiability of low-codimensional subsets in Heisenberg
groups is the almost everywhere existence of suitable approximate tangent
paraboloids.
