This work studies the signal-to-interference-plus-noise-ratio (SINR) meta distribution for the uplink transmission of a Poisson network with Rayleigh fading by using the dominant interferer-based approximation. The proposed approach relies on computing the mix of exact and mean-field analysis of interference. In particular, it requires the distance distribution of the nearest interferer and the conditional average of the rest of the interference. Using the widely studied fractional path-loss inversion power control and modeling the spatial locations of base stations (BSs) by a Poisson point process (PPP), we obtain the meta distribution based on the proposed method and compare it with the traditional beta approximation, as well as the exact results obtained via Monte-Carlo simulations. Our numerical results validate that the proposed method shows good matching and is time competitive.