  We study wall crossings in Bridgeland stability for the Hilbert scheme of
elliptic quartic curves in three dimensional projective space. We provide a
geometric description of each of the moduli spaces we encounter, including when
the second component of this Hilbert scheme appears. Along the way, we prove
that the principal component of this Hilbert scheme is a double blow up with
smooth centers of a Grassmannian, exhibiting a completely different proof of
this known result by Avritzer and Vainsencher. This description allows us to
compute the cone of effective divisors of this component.
