We report a semiconductor saturable absorber mirror (SESAM)-modelocked thin-disk laser oscillator delivering a record 350-W average output power with 940-fs, 39-µJ pulses at 8.88-MHz repetition rate and 37-MW peak power. This oscillator is based on the Yb:YAG gain material and has a large pump spot on the disk. The cavity design includes an imaging scheme, which results in multiple reflections on the disk gain medium to enable a larger output coupling rate compared to those used in thin-disk oscillators with a single reflection on the disk. This reduces the intracavity power for a given output power, thus decreasing the stress on the intracavity components. We operate the laser in a low-pressure environment in order to limit the disk's thermal lensing and drastically reduce the nonlinearity picked up in the intracavity air medium. The combination of the imaging scheme and low-pressure operation paves the way to further power scaling of ultrafast thin-disk oscillators toward the kW milestone.