  We propose an analytical parametrization of the comoving distance and Hubble
parameter to study the cosmic expansion history beyond the vanilla $\Lambda$CDM
model. The parametrization is generalized enough to include the contribution of
spatial curvature and to capture the higher redshift behaviors. With this
parameterization, we study the late time cosmic behavior and put constraints on
the cosmological parameters like present values of Hubble parameter ($H_{0}$),
matter energy density parameter ($\Omega_{m0}$), spatial curvature energy
density parameter ($\Omega_{k0}$) and baryonic matter energy density parameter
($\Omega_{b0}$) using different combinations data like CMB (Cosmic microwave
background), BAO (baryon acoustic oscillation), and SN (Pantheon sample for
type Ia supernovae). We also rigorously study the Hubble tension in the
framework of late time modification from the standard $\Lambda$CDM model. We
find that the late time modification of the cosmic expansion can solve the
Hubble tension between CMB $\&$ SHOES (local distance ladder observation for
$H_{0}$), between CMB+BAO $\&$ SHOES and between CMB+SN $\&$ SHOES, but the
late time modification can not solve the Hubble tension between CMB+BAO+SN and
SHOES. That means CMB, BAO, and SN data combined put strong enough constraints
on $H_{0}$ (even with varying $\Omega_{k0}$) and on other background
cosmological parameters so that the addition of $H_{0}$ prior from SHOES (or
from similar other local distance observations) can not significantly pull the
$H_{0}$ value towards the corresponding SHOES value.
