A graph admitting an automorphism with two orbits of the same length is called a bicirculant. Recently, Jajcay et al. initiated the investigation of the edge-transitive bicirculants with the properties that one of the subgraphs induced by the latter orbits is a cycle and the valence is at least $6$ (Electron. J. Combin., 2019). We show that the complement of the Petersen graph is the only such graph whose order is twice an odd number.