  In this work, we first discuss the possibility that dark energy models with
negative energy density values in the past can alleviate the $H_0$ tension, as
well as the discrepancy with the baryon acoustic oscillation (BAO)
Lyman-$\alpha$ data, both which prevail within the $\Lambda$CDM model. We then
investigate whether two minimal extensions of the $\Lambda$CDM model, together
or separately, can successfully realize such a scenario: (i) the spatial
curvature, which, in the case of spatially closed universe, mimics a negative
density source and (ii) simple-graduated dark energy (gDE), which promotes the
null inertial mass density of the usual vacuum energy to an arbitrary
constant--if negative, the corresponding energy density decreases with redshift
similar to the phantom models, but unlike them crosses below zero at a certain
redshift. We find that, when the Planck data are not included in the
observational analysis, the models with simple-gDE predict interesting and some
significant deviations from the $\Lambda$CDM model. In particular, a spatially
closed universe along with a simple-gDE of positive inertial mass density,
which work in contrast to each other, results in minor improvement to the $H_0$
tension. The joint dataset, including the Planck data, presents no evidence for
a deviation from spatial flatness but almost the same evidence for a
cosmological constant and the simple-gDE with an inertial mass density of order
$\mathcal{O}(10^{-12})\,\rm eV^4$. The latter case predicts almost no deviation
from the $\Lambda$CDM model up until today--so that it results in no
improvement regarding the BAO Ly-$\alpha$ data--except that it slightly
aggravates the $H_0$ tension. We also study via dynamical analysis the history
of the Universe in the models, as the simple-gDE results in futures different
than the de Sitter future of the $\Lambda$CDM model.
