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Aims. Three-dimensional (3D) maps of Galactic interstellar dust are a tool for a wide range of uses. We aim to construct 3D maps of dust extinction in the Local Arm and surrounding regions. Methods. To do this, Gaia EDR3 photometric data were combined with 2MASS measurements to derive extinction towards stars with accurate photometry and relative uncertainties on EDR3 parallaxes of less than 20%. We applied our hierarchical inversion algorithm adapted to inhomogeneous spatial distributions of target stars to this catalogue of individual extinctions. Results. We present the updated 3D dust extinction distribution and provide an estimate of the error on integrated extinctions from the Sun to each area in the 3D map. The full computational area is similar to the one of the previous DR2 map, that is to say with a 6 × 6 × 0.8 kpc 3 volume around the Sun. Due to the addition of fainter target stars, the volume in which the clouds can be reconstructed has increased. Due to the improved accuracy of the parallaxes and photometric data in EDR3, extinctions among neighbouring targets are more consistent, allowing one to reach an increased contrast in the dense areas, while cavity contours are more regular. We show several comparisons with recent results on dust and star distributions. The wavy pattern around the Plane of the dust concentrations is better seen and exists over large regions. Its mean vertical peak-to-peak amplitude is of the order of 300 pc; interestingly, it is similar to the vertical period of the spectacular snail-shaped stellar kinematical pattern discovered in Gaia data. Conclusions. The Gaia EDR3 catalogue allows for a significant improvement of the extinction maps to be made, both in extent and quality. The hierarchical technique confirms its efficiency in the inversion of massive datasets. Future comparisons between 3D maps of interstellar matter and stellar distributions may help to understand which mergers or internal perturbations have shaped the Galaxy within the first 3 kpc.