Abstract: 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 [Propagation of minimality in the supercooled Stefan problem, Christa Cuchiero et al.], 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 [Global solutions to the supercooled Stefan problem with blow-ups: regularity and uniqueness, Delarue et al.]. Here, we prove the uniqueness of physical solutions for oscillatory initial conditions by a new contraction argument that allows to replace the local monotonicity condition of [Global solutions to the supercooled Stefan problem with blow-ups: regularity and uniqueness, Delarue et al.] 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 $1 + W_x- sqrt{2|log|log{x}||}$ 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