Injectors and in particular the positron production systems are a crucial element in linear and circular colliders. The proposed FCC-ee injector linac accelerates both electrons and positrons up to 6 GeV in order to be injected into the Pre-Booster Ring (PBR) with a high intensity of 3.5 nC per bunch for the Z running mode. The current design baseline of the FCC-ee injector linac is based on the SuperKEKB scheme, in which the electron and the positron beams share the same linacs with a fixed target configuration on-axis hole for electron beam passage. One of the main drawbacks of this scheme is the positron production efficiency at the exit of the injector linac. In order to achieve a better performance and a higher positron yield, in this paper we study different bypass injection options. Start-to-end design and optimization for beam production, acceleration and transport from the feeding electron source to the positron damping ring are also being presented.