I'm going to be that guy and say that this is a bad idea. When you're jumping through hoops to get attention, people get used to it. And they start expecting it by default. Especially when it's for a job.
There isn't a playbook for doing something worthy of a company's attention (sans warm intro). But there's something to be said for non-traditional, creative outreach.
However, I am ready to go above and beyond at any position I am at for my own personal reasons, so it won't be much of a stretch, just a taste of what's to come.
Good point, though.