We describe the results of a detailed study of two-dimensional (2D) smectic-C to solid-like phase transition in azobenzene-derivative Langmuir monolayers by the surface pressure vs area-per-molecule isotherm measurement, polarizing optical microscopy, and the grazing incidence synchrotron x-ray diffraction. The phase transition was observed to be weakly first order up to slightly above room temperature with a transition entropy less than 1.0 k(B) per molecule, accompanied by proper characteristics of a nearly ideal horizontal coexistence line on the isotherm, an abrupt change in optical texture, and the emergence of a conspicuous x-ray diffraction peak in the solid-like phase, which indicates a weak positional order with a correlation length of 10-20 nm. Analysis of the x-ray diffraction data within the framework of distorted hexagonal lattice suggests that the solid-like phase may be regarded as a 2D analog of smectic-L phase that is a hexatic smectic phase with the molecules tilted toward a direction between the nearest and the next-nearest bonds. At higher temperatures, the transition became less discontinuous and entirely disappeared above 40 degreesC in all these experimental aspects. We argue that the transition is viewed as a 2D induced hexatic-hexatic transition (analog of smectic-C to smectic-L transition), which has an isolated critical point where the first-order transition comes to an end.