Let us consider a linear operator $T\in L(X)$, with $X$ being an infinite-dimensional separable Banach space. \par It is known that the intersection of an orbit with a closed vector subspace $Y$ can be somewhere dense in $Y$ without being dense in $Y$, see [\textit{S. Grivaux}, Arch. Math. 81, No. 3, 291--299 (2003; Zbl 1056.47007); \textit{R. R. Jiménez-Munguía} et al., J. Math. Anal. Appl. 408, No. 1, 209--212 (2013; Zbl 1306.47010)]. In this work, the author quantifies the set of visiting times of the orbit of a hypercylic vector and shows different results in which the recurrence of the orbits that lay in the subspace is analyzed.