Fan-spine is a special topology in solar atmosphere and is closely related to magnetic null point as well as circular-ribbon flares, which can provide important information for understanding the intrinsic three-dimensional (3D) nature of solar flares. However, the fine structure within the fan has rarely been investigated. In present paper, we investigate a secondary fan-spine (SFS) structure within the fan of a larger fan-spine topology. On 2013 November 18, this large fan-spine structure was traced out due to the partial eruption of a filament, which caused a circular-ribbon flare in active region 11897. The extrapolated 3D magnetic fields and squashing factor Q maps depict distinctly this fan-spine topology, its surrounding quasi-separatrix layer (QSL) halo, and a smaller quasi-circular ribbon with high Q located in the center, which implies the existence of fine structure within the fan. The imaging observations, extrapolated 3D fields, and Q maps on November 17 show that there indeed exists an SFS surrounded by a QSL, which is enveloped by another QSL-halo corresponding to the overlying larger domeshaped fan. Moreover, the material flows caused by the null-point reconnection are also detected along this SFS. After checking the evolution of the underneath magnetic fields, we suggest that the continuous emergence of magnetic flux within the central parasitic region encompassed by the opposite-polarity fields results in the formation of the SFS under the large fan.