We study an emission-dependent dyadic fashion supply chain made up of a supplier and a manufacturer, both of which can reduce their own component/product emissions to serve the carbon-footprint sensitive consumers. With Carbon Tax regulation, we consider four scenarios resulting from two ways in form of adopting transfer price contract and/or introducing third-party emission-reduction service (TPERS) to enhance the efficiency of systematic emission reductions. We refine four models from these corresponding scenarios, which in turn constitute a decision-making framework composed of determining vertical incentives and choosing supply chain structures. By exploiting Stackelberg games in all models, we compare their emission reduction efficiencies and profitability for each pair of settings. Theoretic analysis and numerical studies show that adopting vertical transfer payment schemes can definitely benefit channel carbon footprint reduction and Pareto improvement of supply chain profitability, regardless of whether the emission-reduction service exists or not. However, whether introducing TPERS or not is heavily depending on systematic parameters when the transfer payment incentive is adopted there. We also provide insights on the sensitivity of carbon tax parameters with respect to the supply chain performance, overall carbon emission reduction, vertical incentive and TPERS adopting decision-makings.