We consider the ghost-free higher-order corrections to the Starobinsky model in the old-minimal supergravity, focusing on a sector among several scalar fields in the model that reproduces the scalaron potential in the original Starobinsky model. In general, higher-order corrections cannot be forbidden by symmetries, which likely violate the flatness of the scalaron potential and make inflation difficult in explaining the present Universe. We find a severe constraint on the dimensionless coupling of the R-4 correction as -5.5 x 10(-8) < s < 9.1 x 10(-8) from the recent results of the Planck observation. If we start from the chaotic initial condition, the constraint becomes much more severe. However, in the case in which the coupling of the R-4 correction is positive, the scalaron potential has a local maximum with two local minima at the origin and infinity, which admits topological inflation. In this case, inflation can take place naturally if the coupling satisfies the observational constraints.