Enabled by surface-mediated equilibration, physical vapour deposition can create high-density stable glasses comparable with liquid-quenched glasses aged for millions of years. Deposition is often performed at various rates and temperatures on rigid substrates to control the glass properties. Here we demonstrate that on soft, rubbery substrates, surface-mediated equilibration is enhanced up to 170 nm away from the interface, forming stable glasses with densities up to 2.5% higher than liquid-quenched glasses within 2.5 h of deposition. Gaining similar properties on rigid substrates would require 10 million times slower deposition, taking ~3,000 years. Controlling the modulus of the rubbery substrate provides control over the glass structure and density at constant deposition conditions. These results underscore the significance of substrate elasticity in manipulating the properties of the mobile surface layer and thus the glass structure and properties, allowing access to deeper states of the energy landscape without prohibitively slow deposition rates.