  In this work, we present the Einstein field equations (EFE) in the framework
of a modified matter source and thus find out the solutions for the Wormhole.
To obtain characteristic solutions set, we employ two kinds of equations of
states (EOS), i.e., the van der Waals and polytropic EOS to the EFE. We adopt
embedding class as a general technique for the system and discuss several
physical attributes of the stellar system under embedded wormhole solutions.
Detailed discussion about the matter contents of the thin-shell developed
around the obtained wormhole solutions in the background of Schwarzschild BH
surrounded by cloud and quintessence-type fluid distribution by using a
well-known cut and paste approach. We have also performed stability analysis
using the linearized radial perturbation method for the van der Waals EOS and
polytropic EOS. Several interesting points have evolved from the entire
investigation along with the features of the thin shell around the wormhole.
