  In paper I of this series (Yang et al. 2017, ApJ), we proposed an analytical
method of blind separation ({\bf ABS}) to extract the cosmic magnification
signal in galaxy number distribution and reconstruct the weak lensing power
spectrum. Here we report a new version of the ABS method, with significantly
improved performance. This version is characterized by a shift parameter
$\mathcal{S}$, with the special case of $\mathcal{S}=0$ corresponding to the
original ABS method. We have tested this new version, compared to the previous
one, and confirmed its supreme performance in all investigated situations.
Therefore it supersedes the previous version. The proof of concept studies
presented in this paper demonstrate that it may enable surveys such as LSST and
SKA to reconstruct the lensing power spectrum at $z\simeq 1$ with $1\%$
accuracy. We will test with more realistic simulations to verify its
applicability in real data.
