  Given that the reionization history of cosmic hydrogen is yet to be
stringently constrained, it is worth checking the prospects of doing so using
physically motivated models and available observational data. For this purpose,
we use an extended version of the explicitly photon-conserving semi-numerical
model of reionization, $\texttt{SCRIPT}$, which also includes thermal evolution
of the intergalactic medium (IGM). The model incorporates the effects of
inhomogeneous recombination and radiative feedback self-consistently and is
characterized by five free parameters (two for the redshift-dependent
ionization efficiency, two for the ionizing escape fraction, and another for
reionization temperature increment). We constrain these free parameters by
simultaneously matching with various observational probes, e.g., estimates of
the ionized hydrogen fraction, the CMB scattering optical depth and the galaxy
UV luminosity function. In addition, we include the low-density IGM temperature
measurements obtained from Lyman-$\alpha$ absorption spectra at $z \sim 5.5$, a
probe not commonly used for Bayesian analysis of reionization parameters. We
find that the interplay of the various data sets, particularly inclusion of the
temperature data, leads to tightening of the parameter constraints. Our default
models prefer a late end of reionization (at $z \lesssim 6$), in agreement with
other recent studies. We can also derive constraints on the duration of
reionization, $\Delta z=1.81^{+0.51}_{-0.67}$ and the midpoint of reionization,
$z_{\mathrm{mid}}=7.0^{+0.30}_{-0.40}$. The constraints can be further
tightened by including other available and upcoming data sets.
