Olive-derived biomass is not only a renewable bioenergy resource but also it can be a source of bioproducts, including antioxidants. In this study, the antioxidant composition of extracted olive pomace (EOP) and a new byproduct, the residual fraction from olive pit cleaning (RFOPC or residual pulp) was characterized and compared to olive leafy biomass, which have been extensively studied as a source of antioxidants and other bioactive compounds with pharmacological properties. The chemical characterization showed that these byproducts contain a high amount of extractives; in the case of EOP, it was even higher (52.9%) than in olive leaves (OL) and olive mill leaves (OML) (35.8–45.1%). Then, ultrasound-assisted extraction (UAE) was applied to recover antioxidants from the extractive fraction of these biomasses. The solubilization of antioxidants was much higher for EOP, correlating well with the extractives content and the total extraction yield. Accordingly, this also affected the phenolic richness of the extracts and the differences between all biomasses were diminished. In any case, the phenolic profile and the hydroxytyrosol cluster were different. While OL, OML, and EOP contained mainly hydroxytyrosol derivatives and flavones, RFOPC presented novel trilignols. Other compounds were also characterized, including secoiridoids, hydroxylated fatty acids, triterpenoids, among others, depending on the bioresource. Moreover, after the UAE extraction step, alkaline extraction was applied recovering a liquid and a solid fraction. While the solid fraction could of interest for further valorization as a biofuel, the liquid fraction contained proteins, sugars, and soluble lignin, which conferred antioxidant properties to these extracts, and whose content depended on the biomass and conditions applied.