Yes, in fact in algebra classes in Germany, this is a well-known example or homework problem: Consider the regular action of $G$. Then an element of order $2$ in $G$ is a product of $m$ transpositions, so it is an odd permutation. Thus $G\cap\text{Alt}_{2m}$ is a subgroup of index $2$.

This generalization follows from [Burnside's normal p-complement theorem][1]: Let $p$ be the smallest prime divisor of $\lvert G\rvert$, and suppose that the Sylow $p$-subgroup $P$ is cyclic. Then $G$ has a normal subgroup of index $\lvert P\rvert$.


  [1]: https://en.wikipedia.org/wiki/Normal_p-complement