  We present results of the cross-correlation Ly$\alpha$ intensity mapping with
Subaru/Hyper Suprime-Cam (HSC) ultra-deep narrowband images and Ly$\alpha$
emitters (LAEs) at $z=5.7$ and $6.6$ in a total area of $4$ deg$^2$. Although
overwhelming amount of data quality controls have been performed for the
narrowband images, we further conduct extensive analysis evaluating systematics
of large-scale point-spread-function wings, sky subtractions, and unknown
errors based on physically uncorrelated signals and sources found in real HSC
images and object catalogs, respectively. Removing the systematics, we
carefully calculate cross-correlations between Ly$\alpha$ intensity of the
narrowband images and the LAEs. We tentatively identify very diffuse Ly$\alpha$
emission with the $\simeq 3\sigma$ ($\simeq 2\sigma$) significance at $\gtrsim$
100 comoving kpc (ckpc) far from the LAEs at $z=5.7$ ($6.6$), around and
probably even beyond a virial radius of star-forming galaxies with
$M_\mathrm{h}\sim10^{11}M_\odot$. The diffuse Ly$\alpha$ emission possibly
extends up to $1$,$000$ ckpc with the surface brightness of
$10^{-20}$-$10^{-19}$ erg s$^{-1}$ cm$^{-2}$ arcsec$^{-2}$. We confirm that the
small-scale ($<150$ ckpc) Ly$\alpha$ radial profiles of LAEs are consistent
with those obtained by recent MUSE observations. Comparisons with numerical
simulations suggest that the large-scale ($\sim150$-$1$,$000$ ckpc) Ly$\alpha$
emission are not explained by unresolved faint neighboring galaxies including
satellites, but by a combination of Ly$\alpha$ photons emitted from the central
LAE and other unknown sources, such as cold-gas streams and galactic outflow.
We find no evolution in the Ly$\alpha$ radial profiles of our LAEs from $z=5.7$
to $6.6$, where theoretical models predict a flattening of the profile slope
made by cosmic reionization, albeit with our moderately large observational
errors.
