We present theoretical results at approximate NNLO in QCD for top-quark pair-production total cross sections and top-quark differential distributions at the LHC in the SMEFT. In this work we consider effects of new physics parametrized by the chromomagnetic dipole operator. These approximate results are obtained by adding higher-order soft gluon corrections to the complete NLO calculations. The higher-order corrections are large, and they reduce the scale uncertainties. These improved theoretical predictions can be used to set stronger bounds on top-quark QCD anomalous couplings.