  In the realm of astrophysics, black holes exist within nonvacuum cosmological
backgrounds, making it crucial to investigate how these backgrounds influence
the properties of black holes. In this work, we first introduce a novel static
spherically-symmetric exact solution of Einstein field equations representing a
surrounded hairy black hole. This solution represents a generalization of the
hairy Schwarzschild solution recently derived using the extended gravitational
decoupling method. Then, we discuss how the new induced modification terms
attributed to the primary hairs and various background fields affect the
geodesic motion in comparison to the conventional Schwarzschild case. Although
these modifications may appear insignificant in most cases, we identify
specific conditions where they can be comparable to the Schwarzschild case for
some particular background fields.
