  We study the one-phase one-dimensional supercooled Stefan problem with
oscillatory initial conditions. In this context, the global existence of
so-called physical solutions has been shown recently in [CRSF20], despite the
presence of blow-ups in the freezing rate. On the other hand, for regular
initial conditions, the uniqueness of physical solutions has been established
in [DNS22]. Here, we prove the uniqueness of physical solutions for oscillatory
initial conditions by a new contraction argument that replaces the local
monotonicity condition of [DNS22] with an averaging condition. We verify this
weaker condition for fairly general oscillating probability densities, such as
the ones given by an almost sure trajectory of
$(1+W_x-\sqrt{2x|\log{|\log{x}|}|})_{+}\wedge 1$ near the origin, where $W$ is
a standard Brownian motion. We also permit typical deterministically
constructed oscillating densities, including those of the form
$(1+\sin{1/x})/2$ near the origin. Finally, we provide an example of
oscillating densities for which it is possible to go beyond our main assumption
via further complementary arguments.
