By combining resistive polyhelix and Bitter inserts with a large bore superconducting outsert, the new hybrid magnet in construction at LNCMI-Grenoble will produce in a first step, an overall continuous magnetic field of 43 T in a 34 mm warm bore aperture. After a brief reminder of the specificity of hybrid magnets, namely the strong electromagnetic and mechanical couplings between resistive and superconducting coils, the main specificities of the proposed design are presented. The superconducting coil will provide a nominal magnetic field of 8.5 T in a 1.1 m cold bore diameter relying on the novel development of a Nb-Ti/Cu Rutherford Cable On Conduit Conductor (RCOCC) cooled down to 1.8 K by a bath of superfluid helium at atmospheric pressure. The novelty of the RCOCC concerns the in-laboratory assembly and induction soft-soldering of the Rutherford cable on a Cu-Ag hollow stabilizer allowing a strict control of the interstrand contact resistance. A stainless steel reinforced copper shield inserted between the superconducting and resistive coils will allow reducing the coupling currents induced within the RCOCC as well as the extreme mechanical force exerted on the superconducting coil. After successful thorough reviews of the Grenoble hybrid magnet design anticipating possible upgrades of the maximum magnetic field produced, this project is now well engaged in its construction phase. The status of this project is presented together with the next milestones.