Applied for a job I'm very interested in, is there any way I can increase my chances rather than waiting like usual?

I applied for a certain managerial role in a hospital, but the application process was strange. I was allowed to upload a cover letter, but couldn't change it if I wanted to apply to other jobs at the same hospital, so I kept it blank. Would it help if I contacted a recruiter at this hospital on LinkedIn to politely express my interest in this position, or would that be useless?

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