Let K be a local field of characteristic p and let L/K be a totally ramified Galois extension such that \({{\,\mathrm{Gal}\,}}(L/K)\cong C_{p^n}\). In this paper we find sufficient conditions for L/K to admit a Galois scaffold, as defined in Byott et al (Ann Inst Fourier 68:965–1010, 2018). This leads to sufficient conditions for the ring of integers \({\mathcal {O}}_L\) to be free of rank 1 over its associated order \({\mathfrak {A}}\), and to stricter conditions which imply that \({\mathfrak {A}}\) is a Hopf order in the group ring \(K[C_{p^n}]\).