Assuming you put a decent effort into the process prior to applying:

The point is, at this point you've already been told no. The worst case scenario is that you're told "Still no."

Depending on what it is you do to impress them, that can be used in other scenarios. You make some marketing plan or a brochure or redo a website? Stick it in your back pocket and save it again for later.

So you look desperate? So what? Either they say no, in which case apart from the extra work you've done, you're no further behind, or they say yes and think you're desperate, but you have the job.