In this paper we compute the most general non-diagonal reflection matrices of the RSOS/SOS models and the hard hexagon model using the boundary Yang-Baxter equations. We find a new one-parameter family of the reflection matrices for the RSOS model in addition to the previous result obtained in (Ahn C and Koo W M 1996 Nucl. Phys. B 468 [FS] 461). We also find three classes of the reflection matrices for the SOS model, which has one or two free parameters. For the hard-hexagon model which can be mapped to the RSOS(5) model by folding four RSOS heights into two, the solutions can be obtained similarly with a main difference in the boundary unitarity conditions. Due to this, the reflection matrices can have two free parameters. We show that these extra terms can be identified with the 'decorated' solutions. We also generalize the hard hexagon model by 'folding' the RSOS heights of the general RSOS(p) model and show that they satisfy the integrability conditions such as the Yang-Baxter and boundary Yang-Baxter equations. These models can be solved using the results for the RSOS models.