Simultaneously achieving high piezoresistive sensitivity, stretchability, and good electrical conductivity in conductive elastomer composites (CECs) with carbon nanofillers is crucial for stretchable strain sensor and electrode applications. Here, we report a facile and environmentally friendly strategy to realize these three goals at once by using branched carbon nanotubes, also known as the carbon nanostructure (CNS). Inspired by the brick-wall structure, a robust segregated conductive network of a CNS is formed in the thermoplastic polyurethane (TPU) matrix at a very low filler fraction, which renders the composite very good electrical, mechanical, and piezoresistive properties. An extremely low percolation threshold of 0.06 wt %, currently the lowest for TPU-based CECs, is achieved via this strategy. Meanwhile, the electrical conductivity is up to 1 and 40 S/m for the composites with 0.7 and 4 wt % CNS, respectively. Tunable piezoresistive sensitivity dependent on CNS content is obtained, and the composite with 0.7 wt % filler has a gauge factor up to 6861 at strain ε = 660% (elongation at break is 950%). In addition, this strategy also renders the composites' attractive tensile modulus. The composite with 3 wt % CNS shows 450% improvement in Young's modulus versus neat TPU. This work introduces a facile strategy to fabricate highly stretchable strain sensors by designing CNS network structures, advancing understanding of the effects of polymer-filler interfaces on the mechanical and electrical property enhancements for polymer nanocomposites.