The chiral interaction between light and matter is mainly caused by the spin-momentum locking and makes the chiral quantum optics enter a vigorous development stage. Here, we explore the condition of the perfect chiral interaction between an atom possessing circular dipole and the surface plasmon polariton (SPP) mode. The realization of the perfect chiral interaction must satisfy the following two conditions at the same time. First, the SPP mode should possess the transverse circular polarization; and second, the atom decays mainly into the SPP mode, while the decay through other channel can be ignored. In this paper, we adopt a simple but effective structure to satisfy both of requirements, which is the sandwiched waveguide made of metal. We found that the transverse circular polarization of SPP mode might be achieved within the structure possessing multiple interfaces instead of the interface separating two semi-infinite materials. In our model, the decay rate into SPP mode overwhelms that through traveling wave, which provides higher quantum efficiency. What's more, we found that only the symmetric TM-polarized SPP mode might get the transverse circular polarization. For the sandwiched structure containing metal, the existence of two SPP modes weakens the overall chiral interaction. However, the structure containing left-handed materials (LHMs), which can only support one symmetric TM-polarized SPP mode, can get the nearly perfect chiral interaction. We measure the chiral interaction through the decay rate, radiation field distribution and the unidirectional rate through the energy flux. Our work provides a reference for exploring the perfect chiral interaction in more complex structures and has potential and wide applicability to other optical processes.