Networking for Financial Services Internships

Hello, I am a junior majoring in Economics and Computer Science, and I have been applying to internships in financial services sectors and financial technology industries. I have been invited to several super days, but sadly, I did not receive any offer so I have to apply to other companies. I have not networked with any of the companies that I am applying for right now. Is it a bad move if I send my applications then contact any school alumni who work at the same companies that I find on LinkedIn, and send them messages to polite ask for a phone call or a coffee chat? I am worried that I might come across as annoying or trying too hard -- which will make me get rejected even before getting an interview.

Thanks in advance for all your advices!