  We consider the kinetic derivative nonlinear Schr\"odinger equation, which is
a one-dimensional nonlinear Schr\"odinger equation with a cubic derivative
nonlinear term containing the Hilbert transformation. In our previous work, we
proved small-data global well-posedness of the Cauchy problem on the torus in
Sobolev space $H^s$ for $s>1/2$ by combining the Fourier restriction norm
method with the parabolic smoothing effect, which is available in the periodic
setting. In this article, we improve the regularity range to $s>1/4$ for the
global well-posedness by constructing an effective gauge transformation.
Moreover, we remove the smallness assumption by making use of the dissipative
nature of the equation.
