A modification of the cavity-biased grand canonical Monte Carlo (GCMC) proposed by Mezei is introduced here. Instead of on a fixed grid, test points of cavities are generated at the contact positions around the centers of existing particles. The increase in the probability or bias of finding a cavity is related to the radial distribution function and can hence be corrected. With this new cavity-biased implementation, an improved convergence to equilibrium is demonstrated and higher densities can be attained. Comparisons with the standard GCMC method, and the original cavity-biased scheme of Mezei are made.