  Let $X \subset \mathbb{P}^n$ be a general Fano complete intersection of type
$(d_1,\dots, d_k)$. If at least one $d_i$ is greater than $2$, we show that $X$
contains rational curves of degree $e \leq n$ with balanced normal bundle. If
all $d_i$ are $2$ and $n\geq 2k+1$, we show that $X$ contains rational curves
of degree $e \leq n-1$ with balanced normal bundle. As an application, we prove
a stronger version of the theorem of Z. Tian \cite{Tian}, Q. Chen and Y. Zhu
\cite{ChenZhu} that $X$ is separably rationally connected by exhibiting very
free rational curves in $X$ of optimal degrees.
