Previous positron source designs have assumed that the converter target is significantly wider than the secondary beam size. This paper investigates targets with different transverse profiles to enhance positron production. Starting from the concept of wire-targets (James et al., 1991) proposed in 1991 for the former SLC positron source, we introduce conical-shaped targets to improve the yield by reducing positron reabsorption—an issue exacerbated by high-field solenoid lenses typically used for positron capture. Using Geant4 simulations, we propose new conical targets adapted for the future FCC-ee and its positron test facility P3 (PSI Positron Production experiment) at the Paul Scherrer Institute. Results indicate that conical targets can nearly double the positron production at the target and boost FCC-ee's positron yield by approximately 60%. Additionally, we present thermo-mechanical studies for the conical targets under FCC-ee's primary beam power parameters, and outline the mechanical implementation for a proof-of-principle demonstration at the P3 facility.