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 O(10(-12)) 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.