The force-matching approach to coarse graining, in which the forces that act on site centers are fitted to the respective average forces computed from all-atom molecular dynamics simulations, provides a link between coarse-grained and all-atom molecular dynamics. In the existing implementations, radial site-site interaction potentials are assumed, thus precluding extensive coarse-graining that usually requires anisotropic potentials. In this work, we extended the force-matching approach to coarse-grained models with axially symmetric sites and implemented it to the UNRES model of polypeptide chains developed in our laboratory, in which the only interaction sites are united peptide groups and united side chains, the α-carbon atoms serving as anchor points. The optimizable parameters were those of the UNRES energy function and not whole potential profiles, which provide better transferability. We tested the implementation with the 20-residue tryptophan-cage miniprotein, selected as the training protein, starting from the NEWCT-9P variant of UNRES. The reference forces were obtained from implicit- and explicit-solvent simulations. Using a target function composed of a force-matching term and a maximum-likelihood term that drives the force field at reproducing the NMR-determined conformational ensembles at three selected temperatures, force fields were obtained which did not produce site-site clashes for the structures simulated with all-atom molecular dynamics with AMBER, and modeled the structures of α-helical proteins with resolution comparable to that of the NEWCT-9P force field. The new force fields also produced the free-energy landscapes of tryptophan cage similar to those obtained from the all-atom molecular dynamics runs.