The main theorem of this article provides sufficient conditions for a degree
d
d
finite cover
M
′
M’
of a hyperbolic 3-manifold
M
M
to be a surface bundle. Let
F
F
be an embedded, closed and orientable surface of genus
g
g
, close to a minimal surface in the cover
M
′
M’
, splitting
M
′
M’
into a disjoint union of
q
q
handlebodies and compression bodies. We show that there exists a fiber in the complement of
F
F
provided that
d
d
,
q
q
and
g
g
satisfy some inequality involving an explicit constant
k
k
depending only on the volume and the injectivity radius of
M
M
. In particular, this theorem applies to a Heegaard splitting of a finite covering
M
′
M’
, giving an explicit lower bound for the genus of a strongly irreducible Heegaard splitting of
M
′
M’
. Applying the main theorem to the setting of a circular decomposition associated to a non-trivial homology class of
M
M
gives sufficient conditions for this homology class to correspond to a fibration over the circle. Similar methods also lead to a sufficient condition for an incompressible embedded surface in
M
M
to be a fiber.