  This current analysis offers novel wormhole solutions in the background of
newly developed extended $f(\mathcal{R},\mathcal{L}_m)$ gravity. We use an
anisotropic matter source and a particular type of energy density demonstrating
cold dark matter halo and quantum wave dark matter halo to calculate two
different wormhole solutions. The properties of the exotic matter within the
wormhole geometry and the matter contents via energy conditions are studied in
detail, both analytically and graphically, by showing valid and invalid
regions. The calculated shape functions of wormhole geometry satisfy the
required conditions in both cases. Further, we investigate the stability of the
shell around wormhole structures by considering black hole solutions in the
framework of cold dark matter halo and quantum wave dark matter by assuming
matter contents located at the shell follow the phantom-like equations of
state. Then, for quintessence and phantom energy type equations of state, the
choice of cold dark matter halo has maximum stability at lower equilibrium
shell radii and declines as the radius rises. More petite wormhole throats have
an unstable configuration for the dark energy matter content, while higher
wormhole throat radius has the lowest stability. For the choice of quantum wave
dark matter, the shell around the wormhole structure is unstable for both
quintessence and phantom energy, while dark energy shows a stable configuration
for smaller values of wormhole throat.
