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 $simeq 3sigma$ ($simeq 2sigma$) 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.
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