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We summarize our recent proposals for probing the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><mi>P</mi></mrow></semantics></math></inline-formula>-odd <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>i</mi><mover accent="true"><mi>κ</mi><mo>˜</mo></mover><mover accent="true"><mi>t</mi><mo>¯</mo></mover><msup><mi>γ</mi><mn>5</mn></msup><mi>t</mi><mi>h</mi></mrow></semantics></math></inline-formula> interaction at the LHC and its projected upgrades directly using associated on-shell Higgs boson and top quark or top quark pair production. We first recount how to construct a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><mi>P</mi></mrow></semantics></math></inline-formula>-odd observable based on top quark polarization in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>W</mi><mi>b</mi><mo>→</mo><mi>t</mi><mi>h</mi></mrow></semantics></math></inline-formula> scattering with optimal linear sensitivity to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mi>κ</mi><mo>˜</mo></mover></semantics></math></inline-formula>. For the corresponding hadronic process <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mi>p</mi><mo>→</mo><mi>t</mi><mi>h</mi><mi>j</mi></mrow></semantics></math></inline-formula> we then present a method of extracting the phase-space dependent weight function that allows to retain close to optimal sensitivity to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mi>κ</mi><mo>˜</mo></mover></semantics></math></inline-formula>. For the case of top quark pair production in association with the Higgs boson, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mi>p</mi><mo>→</mo><mi>t</mi><mover accent="true"><mi>t</mi><mo>¯</mo></mover><mi>h</mi></mrow></semantics></math></inline-formula>, with semileptonically decaying tops, we instead show how one can construct manifestly <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><mi>P</mi></mrow></semantics></math></inline-formula>-odd observables that rely solely on measuring the momenta of the Higgs boson and the leptons and <i>b</i>-jets from the decaying tops without having to distinguish the charge of the <i>b</i>-jets. Finally, we introduce machine learning (ML) and non-ML techniques to study the phase-space optimization of such <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><mi>P</mi></mrow></semantics></math></inline-formula>-odd observables. We emphasize a simple optimized linear combination <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="bold">α</mi><mo>·</mo><mi mathvariant="bold">ω</mi></mrow></semantics></math></inline-formula> that gives similar sensitivity as the studied fully fledged ML models. Using <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="bold">α</mi><mo>·</mo><mi mathvariant="bold">ω</mi></mrow></semantics></math></inline-formula> we review sensitivity projections to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mi>κ</mi><mo>˜</mo></mover></semantics></math></inline-formula> at HL-LHC, HE-LHC, and FCC-hh.