Advice: Politely Persistent vs Arrogantly Pushy in Job Rejection Reply

Need advice on follow-up to a rejected job role: I've been eyeing a start-up for 3 yrs that I know I could contribute immensely to given my specialized skill set in digitizing supply chains. I was able to connect to two team members earlier this year who indicated they would be growing in the fall and to keep checking back since I was a good fit. I saw a job role posted on their website and my experience not only matched, but exceed the the description. When I got in contact with the team lead hiring for the role I was told they were impressed with my resume but had more qualified applicants and would not be proceeding with me. I really feel there are few people who would thrive in the role like me and I'm tempted to send a politely persistent follow-up reply with a few solutions/partnerships I could bring to the role and help the company succeed. I dont want to come off arrogant and pushy though, any advice on how this could be worded and if you would appreciate a reply like that?