Another risk you can run into if you 'jump' on someone who is just ending a relationship is that they may (intentionally or not) use you as a
rebound, which may be doomed from the start, if the relationship isn't based so much on liking
you as it is avoiding the hurt of breaking up with the last person (which can hurt you and make you feel like nothing more than a temporary, convenient distraction, in the end).
Yeah, most people aren't in a position to start fresh immediately after breaking up. It hurts and may take time to adjust, and those emotions don't simply vanish.