We present intermediate results from a long-term program of mapping the neutral absorption characteristics of the local interstellar medium, motivated by the availability of accurate and consistent parallaxes from the Hipparcos satellite. Equivalent widths of the interstellar NaI D-line doublet at 5890{AA} are presented for the lines-of-sight towards some 311 new target stars lying within ~350pc of the Sun. Using these data, together with NaI absorption measurements towards a further ~240 nearby targets published in the literature (for many of them, in the directions of molecular clouds), and the ~450 lines-of-sight already presented by (Sfeir et al., 1999, Cat. <J/A+A/346/785>), we show 3D absorption maps of the local distribution of neutral gas towards 1005 sight-lines with Hipparcos distances as viewed from a variety of different galactic projections. The data are synthesized by means of two complementary methods, (i) by mapping of iso-equivalent width contours, and (ii) by density distribution calculation from the inversion of column-densities, a method devised by Vergely et al. (2001A&A...366.1016V). Our present data confirms the view that the local cavity is deficient in cold and neutral interstellar gas. The closest dense and cold gas "wall", in the first quadrant, is at ~55-60pc. There are a few isolated clouds at closer distance, if the detected absorption is not produced by circumstellar material. The observations were obtained during 11 observing runs over the period 1998-2000.