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In the work, we present investigation on decoupling gravitational sources under the framework of $f(R,T)$ gravity. Basically the complete geometric deformation technique has been employed here which facilitates finding exact solutions to the anisotropic astrophysical system smoothly without imposing any particular ansatz for deformation function. Along with this we have also used 5-dimensional Euclidean spacetime in order to describe the embedding Class I spacetime for getting a solvable spherical physical system. The solutions thus obtained show physically interesting as well as viable with new possibilities to sought for. Especially, from the present investigation it is worthy to note that the mixture $f(R,T)$ + CGD translate the scenario beyond the pure GR realm and hence clearly helps to enhance the features of the interior astrophysical aspects of compact stellar objects. Therefore to check the physical acceptability and stability of the stellar system based on the obtained solutions, we have performed a few physical tests which satisfy all the stability criteria, including nonsingular nature of the density as well as pressure.