The use of natural fibers as reinforcement in polymer composite materials is increasing owing to the ecocompatibility. The fabrication of natural fiber-based composite is now an emerging field of study, and it is the most desired choice not only because of its superior qualities such as light weight, stiffness, and low density but also because of being economical and renewable. However, the mechanical characteristics of these composites are a major concern and thus require some additives in the form of nanoclay, which can readily act as a filler material, thereby enhancing the tensile, flexural, and impact strengths of the natural fiber composites. In this regard, a hybrid polymer composite with jute and E-glass fiber as reinforcements, along with Cloisite 20 nanoclay filler material, is fabricated by a hand layup process in a steel mold box and cut to the ASTM standards. The composition of the reinforcements is fixed based on the initial experimental studies and review of the literature findings. The wt.% of jute is fixed in the range of 10 wt.% to 20 wt.% with 5 wt.% interval for each composition, while the wt.% of E-glass fiber varied from 10 wt.% to 30 wt.% with 10 wt.% interval and that of the Cloisite 20 nanoclay filler varied from 2 wt.% to 6 wt.%. The fabricated composite specimens with varying wt.% of reinforcements and nanofiller material are subjected to standard tests for evaluating their mechanical characteristics viz., the tensile, flexural, and impact strength and the morphological characteristics. The results of the experiments have revealed that combining natural and synthetic fibers in a composite increases the impact, tensile, and flexural strength of the material. That is, the composite specimens fabricated with 20 wt.% of jute fiber and 20 wt.% of E-glass fiber and 2 wt.% of Cloisite 20 nanoclay have the maximum tensile strength of 69.7 MPa, tensile modulus of 3816.43 MPa, and impact strength of 178.62 J/m2 and flexural modulus of 275.15 MPa among all the specimens, while the composite specimen fabricated with 20 wt.% of jute fiber and 30 wt.% of E-glass fiber and 4 wt.% of the filler material possesses the maximum flexural strength of 90.22 MPa. This is also ascertained from Taguchi’s optimization studies and statistical model (regression equations) developed.