  Inspired by Schmidt's work on twisted cubics, we study wall crossings in
Bridgeland stability, starting with the Hilbert scheme
$\mathrm{Hilb}^{2m+2}(\mathbb{P}^3)$ parametrizing pairs of skew lines and
plane conics union a point. We find two walls. Each wall crossing corresponds
to a contraction of a divisor in the moduli space and the contracted space
remains smooth. Building on work by Chen--Coskun--Nollet we moreover prove that
the contractions are $K$-negative extremal in the sense of Mori theory and so
the moduli spaces are projective.
