Let $(C,\alpha)$ and $(H,\beta)$ be Hom-bialgebras and $\omega:C\otimes H\rightarrow H\otimes C$ a linear map. We introduce the concept of a Hom-$\omega$-crossed coproduct $(C_{\omega}\bowtie_{\sigma} H,\gamma)$ and we give necessary and sufficient conditions for the new object to be a Hom-Hopf algebra.
