Enhancing the vulcanization rate of elastomers is a common objective for optimizing their performance. Styrene–butadiene rubber (SBR) was vulcanized employing several conventional systems designed with various amounts of sulfur and activators/accelerators. Herein, it was an attempt to synthesize a copolymer from glycidyl methacrylate (GMA) and diethylaminoethyl methacrylate (DEAEMA)-(GMA-co-DEAEMA) and study its effect as a secondary accelerator on the curing characteristics and vulcanizate properties of SBR. The copolymer GMA-co-DEAEMA (III) was grafted with three different functional groups (R) including thioglycolic acid, 2-aminothiophenol, and 2-amino-5-mercaptothiazole, to yield III/S/1, IIII/S/2, and III/S/3 copolymers, respectively. The three-modified copolymers with an added conventional accelerator N-cyclohexyl-2-benzothiazole sulfenamide (CBS) were used to accelerate the efficiency of the vulcanizing agent (sulfur). Additional characterization of the synthesized copolymer and the obtained cured elastomer were carried out through different techniques, including mass and Fourier-transform infrared spectroscopy and thermogravimetric analysis (TGA), in addition to the rheological and mechanical studies. The results revealed that all the subject copolymers, when applied as a secondary accelerator for the SBR vulcanization reaction, enhanced the rate and the state of the cross-linking process. The findings of this study eruditely recommended that large rubber articles, like tires, could be cured more efficiently and cost-effectively when employing the presented copolymer as a secondary accelerator.