We report on an improved ytterbium-doped yttrium aluminum garnet thin-disk multi-pass amplifier for kilowatt-level ultrafast lasers, showcasing excellent beam quality. At a repetition rate of 800 kHz, the 6.8 ps, 276 W seed laser is amplified up to an average power of 1075 W, corresponding to a pulse energy of 1.34 mJ. The 36-pass amplifier is designed as a compact mirror array in which the beam alternately propagates between the mirrors and the disk by a quasi-collimated state. We adopted a quasi-collimated propagation to confine stray and diffracted light by the slight curvature of the disk, which enables us to achieve an outstanding extraction efficiency of up to 57% with excellent beam quality in stable laser operation at high power. The beam quality at 1075 W was measured to be M2 < 1.51. Furthermore, stability testing was demonstrated with a root-mean-square power fluctuation of less than 1.67% for 10 min.