  The rising concern in the Hubble constant tension ($H_0$ tension) of the
cosmological models motivates the scientific community to search for
alternative cosmological scenarios that could resolve the $H_0$ tension. In
this regard, we aim to work on a torsion-based modified theory of gravity which
is an alternative description to the coherence model. We solve numerically for
the Hubble parameter using two exponential Lagrangian functions of torsion $T$
and a trace of energy-momentum tensor $\mathcal{T}$ for the dust case. Further,
we constrain the cosmological and model parameters; to do that, we use Hubble,
SNe Ia, Baryon Acoustic Oscillations, Cosmic Microwave Background samples, and
Markov Chain Monte Carlo (MCMC) simulation through Bayesian statistics. We
obtain the values of Hubble constant $H_0$ for our model, and the outputs align
with the recent observational measurements of $H_0$. In addition, we check the
deviation of our results from model-independent measurements of $H_0$ from
Planck2018, S$H_0$ES, and $H_0$LiCOW experiments. In contrast, our finding
partially solved the $H_0$ tension but gave a new possible direction to
alleviate the $H_0$ tension.
