In the frame of efforts to add value to the Mediterranean currant cultivation and processing sectors, which is essential for their sustainability, sweet wine production is proposed from the finishing side-stream (FSS) of premium quality Corinthian currants, involving complete fermentation using an alcohol-tolerant yeast followed by (i) the addition of FSS to extract sugars or (ii) syrup made from FSS to adjust sweetness. Wine was also made by (iii) ceasing fermentation at the desired sugar level by ethanol addition. The non-fortified wines had 15.2–15.5% ethanol, 115–145 g/L residual sugar, 7.2–7.6 g/L titratable acidity, low volatile acidity (VA; <0.33 g/L), 280–330 mg/L phenolic content (TPC) (as gallic acid), and 23.8–35.6 mg/L antioxidant capacity (AC) (as ascorbic acid). In total, 160 volatiles were identified by SPME GC-MS, including compounds derived from the grapes, the raisin drying, and the fermentation process. The non-fortified wines had better characteristics (mainly VA, AC, and TPC) than the fortified wine, while sweetness adjustment by FSS is the simplest and lowest cost method since it does not involve ethanol or syrup addition. The proposed methods can lead to good quality sweet wines with a characteristic fruity (grape/raisin) flavor that could be commercialized as specialty raisin beverages or liqueurs.