For pt.I see J. Phys. C, vol.21, p.1953 (1988). The present paper is concerned with the problem of the polaronic states of positively charged particles in crystals. In contrast to the commonly considered case of electronic polarons such particles tend to reside on the interstitial sites of the crystal. To work out the consequences of this fact the interstitial-polaron model is investigated. This model is a slight modification of Holstein's molecular-crystal model treated in part I of this series. Using the variational approach introduced in part I of a new class of polaronic states appears to become stable in some parameter region in addition to the various types of states existing for the molecular-crystal model. These states will be called two-site states since they describe a particle effectively tunnelling between two neighbouring sites. They are shown to be increasingly important if a configuration dependence of the transfer integral is taken into account. The two-site states may be considered as a one-dimensional analogue of the 4T states which have been proposed for hydrogen and its isotopes in BCC metals.