Conductive elastomer composites (CECs) with high elasticity, flexibility and sealability of elastomers and the high electrical conductivity of conductive fillers have been widely used in electromagnetic interference shielding, microwave absorpting, stretchable electronics and sensors. To achieve high electrical conductivity, the commonly used fillers are metal powers, metal-coated inorganic particles and carbon black (CB). High filler content is usually required to effectively improve the conductivity, resulting in low elasticity, low flexibility, and poor processability, all of which limit the applications of CECs. Earlier works reported the synergistic effect of two kinds of fillers with different geometrical dimensions, among which CB and carbon nanotubes (CNTs) are two promising candidate that have been widely studied. However, CNTs are easily entangled and aggregated in elastomer matrix with low viscosity, resulting in unsatisfactory conductivity (mostly 104–105 Ω cm). Moreover, little attention has been paid on CECs with high elasticity, high electrical conductivity stability filled with low filler contents (low cost) for stretchable electronics and strain sensors. The relationship between conductive network and electrical conductivity as well as electrical conductivity stability is still not clear. In this study, we designed and prepared CB/CNTs/polymethylvinylsiloxane (PMVS) composites with good elasticity, high electrical conductivity and good electrical conductivity stability by tailoring the filler network structure of the composites with a low content of CB and CNTs in PMVS matrix. The composite with 1.8 vol% of CNTs and 1.2 vol% of CB showed high elasticity, low volume resistivity (271 Ω cm), and high electrical conductivity stability. CNTs used in this study were carbon nanotube arrays (CNTA) that were in situ synthesized on single-layered double hydroxide (LDH) nanosheets by chemical vapor deposition method. The negatively charged CNTs with a loose structure showed no physical entanglement, which can be well dissociated into many single CNTs and dispersed uniformly in PMVS matrix during mechanical shearing. The high aspect ratio of CNTs can help to obtain high conductivity at low filler content, whereas the isotropy of CB helps to rebuild up the new conductive network after tensile strain. On the other hand, these CNTs are curved in the elastomer matrix, become oriented during stretching, and can curve again after recovery, and thus act as nanosprings. The high elasticity of the CNTs nanosprings leads to high elasticity of the PMVS-CB-CNTs composites. We further studied the relationship between conductive network and electrical conductivity as well as electrical conductivity stability of PMVS-CB-CNTs composites. The results showed that CB and CNTs can form a strong dual conductive network at low filler contents, leading to a high conductivity. When stretching, CNTs act as bridges to connect CB and CB aggregations together, maintaining the strong conductive network and thus high electrical conductivity stability under tensile strain. After tensile recovery, CB and CNTs can reform a strong conductive network, leading to high electrical stability during tensile-release cycles.