We consider $Ntimes N$ Hermitian random matrices $H$ consisting of blocks of size $Mgeq N^{6/7}$. The matrix elements are i.i.d. within the blocks, close to a Gaussian in the four moment matching sense, but their distribution varies from block to block to form a block-band structure, with an essential band width $M$. We show that the entries of the Greens function $G(z)=(H-z)^{-1}$ satisfy the local semicircle law with spectral parameter $z=E+mathbf{i}eta$ down to the real axis for any $eta gg N^{-1}$, using a combination of the supersymmetry method inspired by cite{Sh2014} and the Greens function comparison strategy. Previous estimates were valid only for $etagg M^{-1}$. The new estimate also implies that the eigenvectors in the middle of the spectrum are fully delocalized.
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