I believe the answer is no.
If I am reading the article below by Méo correctly, he shows that if $\pi : Y \to X$ is the blowup of the polydisc $X=D^m$ along $Z := \{ z_1 = \dots = z_k = 0 \}$, and if $k \leq p \leq n-2$, then there exists a closed positive $(p,p)$-current $T$ on $X$ such that $T' := \pi^* (T|_{X\setminus Z})$ does not have locally finite mass near $\pi^{-1}(Z)$. If such $S$ exists, then since $\pi$ is a biholomorphism on $Y' := \pi^{-1}(X\setminus Z)$, one would have $S|_{Y'} = T'$, which would thus have locally finite mass.
Reference
Michel Méo, "Image inverse d'un courant positif fermé par une application analytique surjective", C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), no. 12, 1141–1144, MR1396655, Zbl 0858.32012.