A definite treatment of angle quantization, including a discussion of the extended literature on this problem, is given in
Quantization of the canonically conjugate pair angle and orbital angular momentum,
Physical Review A 73.5 (2006): 052104.
In the main text it is mentioned that
There are two typical (generic) examples where the unit circle $S^1$,
parametrized by the momentum angle φ ∈ R mod 2π, represents the
configuration space, whereas the canonically conjugate momentum
variable $p_φ$ represents is either a positive real number $p_φ > 0$,
i.e. $p_φ ∈ R_+$, or a real number, i.e. $p_φ ∈ R$.
The first case corresponds to the question posed in the OP and is treated in fuller detail in another paper of the author (https://arxiv.org/abs/quant-ph/0307069), the second case corresponds to the situation explicitly mentioned in the title of the paper.
For both cases, the problem and its resolution is well captured in the first part of the abstract of the paper:
The question how to quantize a classical system where an angle φ is
one of the basic canonical variables has been controversial since the
early days of quantum mechanics. The problem is that the angle is a
multivalued or discontinuous variable on the corresponding phase
space. The remedy is to replace φ by the smooth periodic functions cos
φ and sin φ.