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 \[[Kollár 1996](https://doi.org/10.1007/978-3-662-03276-3), Thm. 5.14 and Rem. 5.15\]. The reason is that you need certain lower bounds on dimensions of deformation spaces; see \[[Kollár 1996](https://doi.org/10.1007/978-3-662-03276-3), Thm. 1.3\].

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][1]. mr: [1440180][2].

  [1]: https://doi.org/10.1007/978-3-662-03276-3
  [2]: https://mathscinet.ams.org/mathscinet-getitem?mr=1440180