The authors study periodic perturbations of differential equations possessing a homoclinic orbit along which the tangent spaces of the corresponding stable and unstable manifolds intersect in a three-dimensional space. This paper can be seen as a continuation of the work ``Multiple transverse homoclinic solutions near a degenerate homoclinic orbit'' [\textit{X.-B. Lin} et al., J. Differ. Equations 259, No. 1, 1--24 (2015; Zbl 1332.34081)]. As there, the authors derive the bifurcation equations by applying a functional analytical approach. Discussing the bifurcation equations it is shown that up to eight homoclinic orbits arise through saddle-node bifurcations. Also here the method of codiagonalization plays a key role in the discussion of the bifurcation equations.