Non-Saccharomyces yeasts gained an increased interest in winemaking during the last decades, due to their ability to produce relevant amounts of polysaccharides. Moreover, a significant release of glutathione into the wine during fermentation was also observed with these strains, as well as an improvement of color stability and wine aroma profile. Valuable results have been obtained by Hanseniaspora spp. concerning the release of polysaccharides and the production of acetic esters, mainly during fermentation. However, few studies have been carried out about the contribution of this strain on color evolution, protection against oxidation, chemical composition, and aroma profile of wines during aging on lees. Concerning methods for accelerating autolysis, interesting results have been obtained by applying ultrasounds, whereas the possibility to produce yeast extracts (e.g., mannoproteins) or derivatives by means high hydrostatic pressure (HHP) has been already reported. Nevertheless, few is known about the use of this technology for processing fermentation lees. This study aimed to assess the effect of two different fermentation lees subjected to several treatments on improving the chemical composition, volatile profile, and sensory characteristics of a white wine during aging. Lees from single (only S. cerevisiae) and sequential fermentation (H. uvarum followed by S. cerevisiae) were subjected to enzyme addition (commercial preparation with β-glucanase activity) or high hydrostatic pressure, whereas untreated lees were used as control. Treated and untreated lees were added to a white wine (blend of Sauvignon Blanc and Pinot Gris, harvest 2021). The wine samples without lees addition were used as reference. The bottles were stored at 20°C + 1 for three months and lees were weekly resuspended to simulate bâtonnage. The analytical determinations mainly concerned wine color, browning assay, content of glutathione, riboflavin, polysaccharides, wine volatile profile, and sensory evaluation. The addition of lees processed by high hydrostatic pressure, especially those obtained from sequential fermentation, determined an improved color evolution, the highest release of glutathione into the wines (about 7 mg/L), and the best protection against oxidation. Moreover, riboflavin content in these aged wines was basically lower (86 μg/L) compared to those aged on untreated lees (100 μg/L) or on the same lees’ typology subjected to enzyme addition (93 μg/L). Wines aged on lees from sequential fermentation treated by HHP also showed the greatest mean concentrations of polysaccharides (169 mg/L) and an overall improvement of volatile profile, with a generally higher concentration of acetic esters. These aged wines were the most preferred by the tasters during sensory evaluation, obtaining the highest scores for floral and fruity notes. The results of the present study may offer insights to improve wine quality, both modulating lees composition (i.e., using non-Saccharomyces spp. as fermentation co-starters) and applying suitable processing technology for accelerating yeast autolysis. High hydrostatic pressure seems to be a suitable strategy for managing and reducing the duration of wine aging on lees, especially when associated with lees obtained from fermentation carried out with specific yeast strains. The antioxidant properties of lees may be thus improved by integrating biotechnologies and emerging technologies; this might reduce the amount of sulfur dioxide, with an innovative and sustainable approach.