Stars form in cold molecular clouds. However, molecular gas is difficult to observe because the most abundant molecule (H_2_) lacks a permanent dipole moment. Rotational transitions of CO are often used as a tracer of H_2_, but CO is much less abundant and the conversion from CO intensity to H_2_ mass is often highly uncertain. Here we present a new method for estimating the column density of cold molecular gas ({Sigma}_gas_) using optical spectroscopy. We utilize the spatially resolved H{alpha} maps of flux and velocity dispersion from the Sydney-AAO Multi-object Integral field spectrograph (SAMI) Galaxy Survey. We derive maps of {Sigma}_gas_ by inverting the multi-freefall star formation relation, which connects the star formation rate surface density ({Sigma}_SFR_) with {Sigma}_gas_ and the turbulent Mach number (M). Based on the measured range of {Sigma}_SFR_=0.005-1.5M_{sun}_/yr/kpc^2^ and M=18-130, we predict {Sigma}_gas_=7-200M_{sun}_/pc^2^ in the star-forming regions of our sample of 260 SAMI galaxies. These values are close to previously measured {Sigma}_gas_ obtained directly with unresolved CO observations of similar galaxies at low redshift. We classify each galaxy in our sample as 'star-forming' (219) or 'composite/AGN/shock' (41), and find that in 'composite/AGN/shock' galaxies the average {Sigma}_SFR_, M and {Sigma}_gas_ are enhanced by factors of 2.0, 1.6 and 1.3, respectively, compared to star-forming galaxies. We compare our predictions of {Sigma}_gas_ with those obtained by inverting the Kennicutt-Schmidt relation and find that our new method is a factor of 2 more accurate in predicting {Sigma}_gas_, with an average deviation of 32 per cent from the actual {Sigma}_gas_.