Band crossings observed in a wide range of condensed matter systems are recognized as a key to understand low-energy fermionic excitations that behave as massless Dirac particles. Despite rapid progress in this field, the exploration of non-equilibrium topological states remains scarce and it has potential ability of providing a new platform to create unexpected massless Dirac states. Here we show that in a cw-laser driven semiconductor quantum-well, the optical Stark effect conducts bulk-band crossing, and the resulting Floquet-Dirac semimetallic phase supports an unconventional edge state in the projected one-dimensional Brillouin zone. Further, we reveal that this edge state mediates a transition between topological and non-topological edge states that is caused by tuning the laser intensity. The existence of the respective edge states and the related topological numbers are understood in a unified manner in terms of the laser-induced polarization reflecting parity hybridization in the bulk Brillouin zone.