Current gravitational-wave detectors rely on the use of Michelson interferometers. One crucial limitation of their sensitivity is the thermal noise of their optical components. Thus, for example, fluctuational deformations of the mirror surface are probed by a laser beam being reflected from the mirrors at normal incidence. Thermal noise models are well evolved for that case but mainly restricted to single reflections. In this work, we present the effect of two consecutive reflections under a non-normal incidence onto mirror thermal noise. This situation is inherent to detectors using a geometrical folding scheme such as GEO 600. We revise in detail the conventional direct noise analysis scheme to the situation of non-normal incidence allowing for a modified weighting function of mirror fluctuations. An application of these results to the GEO 600 folding mirror for Brownian, thermoelastic, and thermorefractive noise yields an increase of displacement noise amplitude by 20% for most noise processes. The amplitude of thermoelastic substrate noise is increased by a factor of 4 due to the modified weighting function. Thus, the consideration of the correct weighting scheme can drastically alter the noise predictions and demands special care in any thermal noise design process.