We study thermodynamic properties of N-f = 2 + 1 QCD on the lattice adopting a nonperturbatively Ooa+-improved Wilson quark action and the renormalization group-improved Iwasaki gauge action. To cope with the problems due to explicit violation of the Poincare and chiral symmetries, we apply the small flow-time expansion (SFtX) method based on the gradient flow, which is a general method to correctly calculate any renormalized observables on the lattice. In this method, the matching coefficients in front of operators in the small flow-time expansion are calculated by perturbation theory thanks to the asymptotic freedom around the small flow-time limit. In a previous study using one-loop matching coefficients, we found that the SFtX method works well for the equation of state extracted from diagonal components of the energy-momentum tensor and for the chiral condensates and susceptibilities. In this paper, we study the effect of two-loop matching coefficients which have been calculated by Harlander et al. recently. We also test the influence of the renormalization scale in the SFtX method. We find that, by adopting the mu(0) renormalization scale of Harlander et al. instead of the conventional mu(d) = 1/root 8t scale, the linear behavior at large flow-times is improved so that we can perform the t -> 0 extrapolation of the SFtX method more confidently. In the calculation of the two-loop matching coefficients by Harlander et al., the equation of motion for quark fields was used. For the entropy density in which the equation ofmotion has no effects, we find that the results using the two-loop coefficients agree well with those using one-loop coefficients. On the other hand, for the trace anomaly which is affected by the equation of motion, we find discrepancies between the one- and two-loop results at high temperatures. By comparing the results of one-loop coefficients with and without using the equation of motion, the main origin of the discrepancies is suggested to be attributed to contamination of O((aT)(2)) = O(1/N-t(2)) discretization errors in the equation of motion at N-t less than or similar to 10.