This paper presents multiplicity measurements of charged hadrons produced in 120 GeV/$c$ proton-carbon interactions. The measurements were made using data collected at the NA61/SHINE experiment during two different data-taking periods, with increased phase space coverage in the second configuration due to the addition of new subdetectors. Particle identification via $dE/dx$ was employed to obtain double-differential production multiplicities of $π^+$, $π^-$, $p$, $\bar{p}$, $K^+$ and $K^-$. These measurements are presented as a function of laboratory momentum in intervals of laboratory polar angle covering the range from 0 to 450 mrad. They provide crucial inputs for current and future long-baseline neutrino experiments, where they are used to estimate the initial neutrino flux.