It is always a challenge how to overcome speckle noise interference in the phase reconstruction for coherent digital holography (CDH) and its application, as this issue has not been solved well so far. In this paper, we are proposing an enhanced anti-speckle deep neural unwrapping network (E-ASDNUN) approach to achieve high quality of absolute phase reconstruction for CDH. The method designs a special network-based noise filter and embeds it into a deep neural unwrapping network to enhance anti-noise capacity in the image feature recognition and extraction process. The numerical simulation and experimental test on the phase unwrapping reconstruction and the image quality evaluation under the noise circumstances show that the E-ASDNUN approach is very effective against the speckle noise in realizing the high quality of absolute phase reconstruction. Meanwhile, it also demonstrates much better robustness than the typical U-net neural network and the traditional phase unwrapping algorithms in reconstructing high wrapping densities and high noise levels of phase images. The E-ASDNUN approach is also examined and confirmed by measuring the same phase object using a commercial white light interferometry as a reference. The result is perfectly consistent with that obtained by the E-ASDNUN approach.