  We consider cosmological dynamics in the theory of gravity with the scalar
field possessing the nonminimal kinetic coupling to curvature given as $\kappa
G^{\mu\nu}\phi_{,\mu}\phi_{,\nu}$, and the Higgs-like potential
$V(\phi)=\frac{\lambda}{4}(\phi^2-\phi_0^2)^2$. Using the dynamical system
method, we analyze stationary points, their stability, and all possible
asymptotical regimes of the model under consideration. We show that the Higgs
field with the kinetic coupling provides an existence of accelerated regimes of
the Universe evolution. There are three possible cosmological scenarios with
acceleration: (i) {\em The late-time inflation} when the Hubble parameter tends
to the constant value, $H(t)\to H_\infty=(\frac23 \pi G\lambda\phi_0^4)^{1/2}$
as $t\to\infty$, while the scalar field tends to zero, $\phi(t)\to 0$, so that
the Higgs potential reaches its local maximum $V(0)=\frac14 \lambda\phi_0^4$.
(ii) {\em The Big Rip} when $H(t)\sim(t_*-t)^{-1}\to\infty$ and
$\phi(t)\sim(t_*-t)^{-2}\to\infty$ as $t\to t_*$. (iii) {\em The Little Rip}
when $H(t)\sim t^{1/2}\to\infty$ and $\phi(t)\sim t^{1/4}\to\infty$ as
$t\to\infty$. Also, we derive modified slow-roll conditions for the Higgs field
and demonstrate that they lead to the Little Rip scenario.
