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New results for the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>T</mi><mn>20</mn></msub></semantics></math></inline-formula>-component of the tensor-analyzing power of the incoherent negative pion photoproduction are presented. The experiment was performed for the electron beam energy of 800 MeV at the VEPP-3 storage ring in 2021. To extract the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>T</mi><mn>20</mn></msub></semantics></math></inline-formula>-component, we used asymmetry with respect to the change in the sign of the tensor polarization of the deuteron target. Identification of the reaction events was carried out by the detection of two protons in coincidence. Experimental data were compared with the results of statistical simulation, considering the interaction between the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mi>N</mi></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>π</mi><mi>N</mi></mrow></semantics></math></inline-formula> subsystems in the final state of the reaction.