If $f$ is a non-constant, entire analytic function, which is odd, then it is also surjective, due to Picard's Little Theorem. Clearly, $f(0)=0$, and hence $0$ is not missed, and if $f(z)\ne a$, for all $z$, then $f(z)\ne-a$, for all $z$. So if $f$ is not onto, it misses at least two values, and Picard's Little Theorem implies that $f$ is constant.
Is it possible to show this without Picard's Little Theorem?
Otherwise, Is it possible to derive Picard's Little Theorem from this odd version of Picard's Little Theorem?