  We investigate emergent topological gapless phases in the square-lattice
Kitaev model with additional hopping terms. In the presence of nearest-neighbor
hopping only, the model is known to exhibit gapless phases with two topological
gapless points. When the strength of the newly added next-nearest-neighbor
hopping is smaller than a certain value, qualitatively the same phase diagram
persists. We find that further increase of the extra hopping results in a new
topological phase with four gapless points. We construct a phase diagram to
clarify the regions of emergent topological gapless phases as well as
topologically trivial ones in the space of the chemical potential and the
next-nearest-neighbor hopping strength. We examine the evolution of the gapless
phases in the energy dispersions of the bulk as the chemical potential varies.
The topological properties of the gapless phases are characterized by the
winding numbers of the present gapless points. We also consider the ribbon
geometry to examine the corresponding topological edge states. It is revealed
that Majorana-fermion edge modes exist as flat bands in topological gapless
phases. We also perform the analytical calculation as to Majorana-fermion
zero-energy modes and discuss its implications on the numerical results.
