<p>The optimal design of dose-finding trials with an active control has been investigated in the literature for regression models that arise from an accurate error distribution. In this paper, we mainly determine optimal designs for estimating the smallest dose achieving the same treatment effect as the active control from the perspective of the second-order least squares estimation (SLSE). This estimation method has been demonstrated to be asymptotically more efficient than the ordinary least squares estimation (OLSE) when the error distribution is asymmetric. More precisely, we develop optimal design theory based on the SLSE, including equivalence theorems for <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\phi _p\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ϕ</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation>-optimal designs and a geometric characterization of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\tilde{c}\)</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi>c</mi> <mo stretchy="false">~</mo> </mover> </math></EquationSource> </InlineEquation>-optimal designs, which are then illustrated with examples. Furthermore, we investigate the finite sample properties of optimal designs and compare the SLSE with OLSE through simulation, numerical results show that the variance reduction of the SLSE is quite significant when optimal designs based on the SLSE are used for certain situations. Finally, the sensitivity of optimal designs to parameter mis-specifications has also been discussed.</p>