Abstract Scaling regularities are examined associated with effect of additive noise on a critical circle map at the golden-mean rotation number. We present an improved numerical estimate for the scaling constant of Hamm and Graham (Phys. Rev. A46, (1992) 6323) responsible for the effect of noise, γ = 2.3061852653 . Decrease of the noise amplitude by this number ensures possibility to distinguish one more level of fractal-like structure, associated with increase of characteristic time scale by the golden mean factor ( 5 + 1 ) / 2 . Numeric results demonstrating evidence of the expected scaling are presented, e.g. portraits of the noisy attractors, devil's staircase plots, Lyapunov charts on the parameter plane in different scales.