An improved e expansion in the d-dimensional (d > 2) stochastic theory of turbulence is constructed at two-loop order, which incorporates the effect of pole singularities at d → 2 in coefficients of the ε expansion of universal quantities. For a proper account of the effect of these singularities, two different approaches to the renormalization of the powerlike correlation function of the random force are analyzed near two dimensions. By direct calculation, it is shown that the approach based on the mere renormalization of the nonlocal correlation function leads to contradictions at two-loop order. On the other hand, a two-loop calculation in the renormalization scheme with the addition to the force correlation function of a local term to be renormalized instead of the nonlocal one yields consistent results in accordance with the ultraviolet (UV) renormalization theory. The latter renormalization prescription is used for the two-loop renormalization-group analysis amended with partial resummation of the pole singularities near two dimensions, leading to a significant improvement of the agreement with experimental results for the Kolmogorov constant.