The improvement of the high-temperature oxidation resistance remains an ambitious goal for the design of new γ/γ′-strengthened Co-base superalloys, since their oxidation resistance beyond 800 °C still ranks behind their Ni-base counterparts. To better understand the origin of the poor oxidation resistance at higher temperatures, this study focuses on early stages of oxidation of four quaternary (Co-Al-W-Ta system) Co-base model alloys with a two-phase γ/γ′-microstructure and varying γ′-volume fraction at 800 °C, 850 °C and 900 °C. Based on time-resolved isothermal gravimetric analysis (TGA) in synthetic air and detailed electron microscopic analysis, the role of the γ-channel width (or γ′-volume fraction), the surface preparation prior to exposure (polishing versus shot-peening), and the heating conditions (synthetic air versus argon) on protective alumina growth is elucidated. Firstly, for alloys of increased γ′-volume fractions slower oxidation kinetics prevailed. Secondly, the two-phase microstructure was found to decisively affect the propagation of the internal oxidation front at the early stages of oxidation. Thirdly, shot-peening prior to exposure together with a lack of oxygen availability during heating was identified to foster protective alumina growth, accompanied by TCP-phase formation in the substrate. The critical role of a high Al availability in the alloy for a rapid growth of protective alumina and the relating challenges in alloy development regarding, for example, phase stability in this relatively novel Co-base alloy class are discussed in detail.