Quiet Sun magnetism plays an important role in the global energy balance of the solar atmosphere. The study of the magnetism of the quiet Sun has been a major effort in the last decades, and, as a result, very important advances in our knowledge have been achieved. The recent commissioning of the Daniel K. Inouye Solar Telescope, the largest ground-based solar telescope in the world today, provides us with data of high spectropolarimetric quality and high spatial resolution. The combination of data acquired with the ViSP instrument on the DKIST telescope around 630.1 nm and multiline inversions enables the achievement of exceptionally high spatial resolution in the solar atmosphere, both at the surface and in depth. In this paper we determine the log(gf) values of the spectral lines around 630.1 nm observed with the ViSP instrument and search for the best multi-inversion strategy, by means of MHD simulations, to analyze quiet Sun magnetism data within such spectral range. We present results of quiet Sun magnetism with special emphasis on its evolution with optical depth. In the internetwork, we find the decrease with height of the averaged magnetic field strength and the zone from which the fields become more horizontal. Additionally, we explore the depth-dependent evolution of the network's canopy, uncovering intriguing insights into its thermodynamic and magnetic properties. Notably, we detect a temperature enhancement in very localized areas of the vicinity of the network and distinguish mass motions with opposing velocities inside the flux tubes.