The bound $-K_X \cdot C \le \dim X + 1$ can be guaranteed if $X$ has local complete intersection singularities, and $f(C)$ intersects the smooth locus of $X$; see [Kollár 1996, Thm. 5.14 and Rem. 5.15]. The reason is that you need certain lower bounds on dimensions of deformation spaces; see [Kollár 1996, Thm. 1.3].
I don't know, however, if there have been improvements since then.
[Kollár 1996] J. Kollár. Rational curves on algebraic varieties. Ergeb. Math. Grenzgeb. (3), Vol. 32. Berlin: Springer-Verlag, 1996. doi: 10.1007/978-3-662-03276-3. mr: 1440180.