We present a multiphase, resolved study of the galactic wind extending from the nearby starburst galaxy NGC 4666. For this, we use VLT/MUSE observations from the GECKOS program and H I data from the WALLABY survey. We identify both ionized and H I gas in a biconical structure extending to at least $z\sim$ 8 kpc from the galaxy disc, with increasing velocity offsets above the mid-plane in both phases, consistent with a multiphase wind. The measured electron density, using [S II], differs significantly from standard expectations of galactic winds. We find electron density declines from the galaxy centre to $\sim 2$ kpc, then rises again, remaining high ($\sim 100-300$ cm$^{-3}$) out to $\sim$5 kpc. We find that H I dominates the mass loading. The total H I mass outflow rate (above $z~>2$ kpc) is between $5-13~{\rm M}_{\odot }~\rm yr^{-1}$, accounting for uncertainties from disc-blurring and group interactions. The total ionized mass outflow rate (traced by H$\alpha$) is between 0.5 and $5~{\rm M}_{\odot }~\rm yr^{-1}$, depending on $n_e(z)$ assumptions. From ALMA/ACA observations, we place an upper limit on CO flux in the outflow which correlates to $\lesssim 2.9~{\rm M}_{\odot }~\rm yr^{-1}$. We also show that the entire outflow is not limited to the bicone, but a secondary starburst at the edge generates a more widespread outflow, which should be included in simulations. The cool gas in NGC 4666 wind has insufficient velocity to escape the halo of a galaxy of its mass, especially because most of the mass is present in the slower atomic phase. This strong biconical wind contributes to gas cycling around the galaxy.