Let F be the lifting of a circle map of degree one. In [R. Bam´on, I. P. Malta, M. J. Pacifico and F.Takens, Ergodic Theory Dyn. Syst. 4, 493-498 (1984; Zbl 0605.58027)], a notion of F-rotation interval of a point x 2 S1 was given. In this paper, we define and study a new notion of a rotation set of a point which preserves more of the dynamical information contained in the sequences {Fn(y)}1n=0 than the one preserved from [R. Bam´on et al., loc. cit.]. In particular, we characterize dynamically the endpoints of these sets and we obtain an analogous version of the main theorem in [Bam´on et al., 1984] in our settings.