We analyze data from two eclipse ingresses of Her X-1 observed with Ginga on 1989 April 30 and May 19. These observations occur, respectively, during the MAIN HIGH and SHORT HIGH states in the 35 day modulation of Her X-1 intensity. We find significant residual X-ray flux during eclipse, with a gradual decrease in flux following the occultation of the neutron star by the atmosphere of HZ Her. During the central part of the eclipse the count rate becomes nearly constant, at 0.5 mCrab in the energy range 1.7-36.8 keV. From a spectral analysis of the residual emission during the total eclipse of the central source in the MAIN HIGH state, we determine the energy spectral index, alpha = 0.8, similar to that before eclipse. A remarkable feature of the eclipse spectrum is that it does not show a significant iron line feature in contrast to massive wind-fed pulsars, such as Vela X-1 and Cen X-3. From a timing analysis of the same eclipse data, we show that there are no pulses. These results imply that the emission comes from the scattering of continuum X-rays by material in a region considerably larger than the companion star. An extended accretion disk corona may be responsible for this scattering. However, partial eclipse of an extended accretion disk corona is insufficient to account for the count rates in mid-eclipse, when known parameters of the binary system are used. Based on the present results, we suggest that scattering occurs not only in the accretion disk corona but also in the circumstellar matter surrounding the system of Her X-1/HZ Her.