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Aims. Classical Cepheids provide the foundation for the empirical extragalactic distance ladder. Milky Way Cepheids are the only stars in this class accessible to trigonometric parallax measurements. However, the parallaxes of Cepheids from the second Gaia data release (GDR2) are affected by systematics because of the absence of chromaticity correction, and occasionally by saturation. Methods. As a proxy for the parallaxes of 36 Galactic Cepheids, we adopt either the GDR2 parallaxes of their spatially resolved companions or the GDR2 parallax of their host open cluster. This novel approach allows us to bypass the systematics on the GDR2 Cepheids parallaxes that is induced by saturation and variability. We adopt a GDR2 parallax zero-point (ZP) of −0.046 mas with an uncertainty of 0.015 mas that covers most of the recent estimates. Results. We present new Galactic calibrations of the Leavitt law in the V, J, H, KS, and Wesenheit WH bands. We compare our results with previous calibrations based on non-Gaia measurements and compute a revised value for the Hubble constant anchored to Milky Way Cepheids. Conclusions. From an initial Hubble constant of 76.18 ± 2.37 km s−1 Mpc−1 based on parallax measurements without Gaia, we derive a revised value by adopting companion and average cluster parallaxes in place of direct Cepheid parallaxes, and we find H0 = 72.8 ± 1.9 (statistical + systematics) ±1.9 (ZP) km s−1 Mpc−1 when all Cepheids are considered and H0 = 73.0 ± 1.9 (statistical + systematics) ±1.9 (ZP) km s−1 Mpc−1 for fundamental mode pulsators only. [ABSTRACT FROM AUTHOR]