Job seeking tip: if you have to write a cover letter, it's a bad job.

The hiring manager will look at *oodles* of resumes and you think your 3-5 paragraphs of prose are going to help at all? Don't think so, straight to the circular file

Good alternative: work every string in your network to find the hiring manager's contact info. A week after you apply, send them an email saying you did apply, here's your attached resume. Would love to go over details and happy to answer any questions they have. 4 sentences max... but, you've shown that you're resourceful and proactive and the manager saw your resume (it may have been screened by HR)