  In the second paper of this series, we developed a new distance determination
method using the median $J$ magnitude of carbon-rich asymptotic giant branch
stars (CS) as standard candles and the Magellanic Clouds as the fundamental
calibrators. The $J$-band CS luminosity function was modeled using a modified
Lorentzian distribution whose parameters were used to determined whether the
LMC or SMC was the most suitable calibrator. In this third paper of the series,
we expand our sample of galaxies and introduce a more robust method to
determine the parameters of the Lorentzian model. The new fitting method uses
an un-binned maximum likelihood estimator to determine the parameters of the
Lorentzian model resulting in parameter errors that are significantly smaller
compared to the second paper. We test our method in NGC 6822, IC 1613, NGC 3109
and WLM. We also estimate the distances to the same sample of galaxies via the
tip of the red giant branch (TRGB) detection method. Our results from the CS
measurements agree well with those obtained from the TRGB.
