  In this paper, we perform the first application of the hybrid method (exact
low modes plus stochastically estimated high modes) for all-to-all propagators
to the HAL QCD method. We calculate the HAL QCD potentials in the $I=2$
$\pi\pi$ scattering in order to see how statistical fluctuations of the
potential behave under the hybrid method. All of the calculations are performed
with the 2+1 flavor gauge configurations on $16^3 \times 32$ lattice at the
lattice spacing $a \approx 0.12$ fm and $m_{\pi} \approx 870$ MeV. It is
revealed that statistical errors for the potential are enhanced by stochastic
noises introduced by the hybrid method, which, however, are shown to be reduced
by increasing the level of dilutions, in particular, that of space dilutions.
From systematic studies, we obtain a guiding principle for a choice of dilution
types/levels and a number of eigenvectors to reduce noise contaminations to the
potential while keeping numerical costs reasonable. We also confirm that we can
obtain the scattering phase shifts for the $I=2$ $\pi\pi$ system by the hybrid
method within a reasonable numerical cost, which are consistent with the result
obtained with the conventional method. The knowledge we obtain in this study
will become useful to investigate hadron resonances which require quark
annihilation diagrams such as the $\rho$ meson by the HAL QCD potential with
the hybrid method.
