<p>Recent advances in the off-shell formulation of the Double Copy (DC) procedure have revealed a profound connection between gauge theories and T-duality invariant frameworks. The main example is Double Field Theory (DFT), emerging as the off-shell DC of Yang-Mills theory up to cubic order in perturbations. Extending this procedure to a higher-derivative gauge theory gives rise to a Higher-Derivative Double Theory (HDDT), which incorporates Weyl gravity along with <i>b</i>-field and dilaton contributions, all in a T-duality invariant manner. In this work, we show that the quadratic contributions of HDDT are directly related (up to field redefinitions) to DFT+, a T-duality invariant model associated with the bosonic string that incorporates first-order <i>α</i>^′ corrections upon parameterization. Our results expand the potential applications of the off-shell DC program towards constructing perturbative <i>α</i>^′-corrected Lagrangians, while also opening up possibilities for reversing the map by considering the single and zeroth copies.</p>