Highly sensitive conductive polymer composites (CPCs) are designed, employing a facile and low-cost extrusion manufacturing process for both low and high strain sensing in the field of e.g. structural health/damage monitoring and human body movement tracking. Focus is on the morphology control for extrusion processed carbon black (CB)-filled CPCs, utilizing binary and ternary composites based on thermoplastic polyurethane (TPU) and olefin block copolymer (OBC). The relevance of the correct CB amount, kinetic control through a variation of the compounding sequence, and thermodynamic control induced by annealing is highlighted, considering a wide range of experimental (e.g. static and dynamic resistance/SEM/rheological measurements) and theoretical analyses. High CB mass fractions (20 m%) are needed for OBC (or TPU)-CB binary composites but only lead to an intermediate sensitivity as their conductive network is fully-packed and therefore difficult to be truly destructed. Annealing is needed to enable a monotonic increase of the relative resistance with respect to strain. With ternary composites a much higher sensitivity with a clearer monotonic increase results provided that a low CB mass fraction (10-16 m%) is used and annealing is applied. In particular, with CB first dispersed in OBC and annealing a less compact, hence, brittle conductive network (10-12 m% CB) is obtained, allowing high performance sensing.