We present extensive new ab initio path integral Monte Carlo (PIMC) simulation results for the chemical potential of the warm dense uniform electron gas (UEG), spanning a broad range of densities and temperatures. This is achieved by following two independent routes, (i) based on the direct estimation of the free energy [Dornheim et al., Phys. Rev. B 111, L041114 (2025)10.1103/PhysRevB.111.L041114] and (ii) using a histogram estimator in PIMC simulations with a varying number of particles. We empirically confirm the expected inverse linear dependence of the exchange-correlation (XC) part of the chemical potential on the simulated number of electrons, which allows for a reliable extrapolation to the thermodynamic limit without the necessity for an additional finite-size correction. We find very good agreement (within Δμxc≲0.5%) with the previous parametrization of the XC-free energy by Groth et al. [Phys. Rev. Lett. 119, 135001 (2017)0031-900710.1103/PhysRevLett.119.135001], which constitutes an important cross validation of current state-of-the-art UEG equations of state. In addition to being interesting in its own right, our study constitutes the basis for the future PIMC based investigation of the chemical potential of real warm dense matter systems starting with hydrogen.