Integrated sensing and communication (ISAC) is recognized as one of the key enabling technologies for sixth-generation (6G) wireless communication networks, facilitating diverse emerging applications and services in an energy and cost-efficient manner. This paper proposes a multi-user multi-target ISAC system to enable full-space coverage for communication and sensing tasks. The proposed system employs a hybrid simultaneous transmission and reflection reconfigurable intelligent surface (STAR-RIS) comprising active transmissive and passive reflective elements. In the proposed scheme, the passive reflective elements support communication and sensing links for local communication users and sensing targets situated within the same physical region as the base station (BS), while low-power active transmissive elements are deployed to improve sensing performance and overcome high path attenuation due to multi-hop transmission for distant communication users and sensing targets situated far from of the coverage area of the BS. Moreover, to optimize the transmissive/reflective coefficients of the hybrid STAR-RIS, a semi-definite relaxation (SDR)-based algorithm is proposed. Furthermore, to evaluate communication and sensing performance, signal-to-interference-noise ratio (SINR) and Cramer-Rao bound (CRB) metrics have been derived and investigated via conducting extensive computer simulations.