  The widely used Milky Way dust reddening map, the Schlegel, Finkbeiner, &
Davis (1998; SFD) map, was found to contain extragalactic large-scale structure
(LSS) imprints (Chiang & M\'enard 2019). Such contamination is inherent in maps
based on infrared emission, which pick up not only Galactic dust but also the
cosmic infrared background (CIB). When SFD is used for extinction correction,
over-correction occurs in a spatially correlated and redshift-dependent manner,
which could impact precision cosmology using galaxy clustering, lensing, and
supernova Ia distances. Similarly, LSS imprints in other Galactic templates can
affect intensity mapping and cosmic microwave background experiments. This
paper presents a generic way to remove LSS traces in Galactic maps and applies
it to SFD. First, we measure descriptive summary statistics of the CIB in SFD
by cross-correlating the map with spectroscopic galaxies and quasars in SDSS
tomographically as functions of redshift and angular scale. To reconstruct the
LSS on the map level, however, additional information on the phases is needed.
We build a large set of 180 overcomplete, full-sky basis template maps from the
density fields of over 600 million galaxies in WISE and find a linear
combination that reproduces all of the high-dimensional tomographic two-point
statistics of the CIB in SFD. After subtracting this reconstructed LSS/CIB
field, the end product is a full-sky Galactic dust reddening map that
supersedes SFD, carrying all Galactic features therein, with maximally
suppressed CIB. We release this new dust map dubbed CSFD, the Corrected SFD, at
https://idv.sinica.edu.tw/ykchiang/CSFD.html and NASA's LAMBDA archive.
