  We study one-dimensional (1D) Floquet topological insulators with chiral
symmetry going beyond the standard rotating wave approximation. The occurrence
of many anticrossings between Floquet replicas leads to a dramatic extension of
phase diagram regions with stable topological edge states (TESs). We present an
explicit construction of all TESs in terms of a truncated Floquet Hamiltonian
in frequency space, prove the bulk-boundary correspondence, and analyze the
stability of the TESs in terms of their localization lengths. We propose
experimental tests of our predictions in curved bilayer graphene.
