And I do 3-4 hiring interviews a week here in NYC. What connections? First show that you can code; a lot don't pass this filter. Then show that you can architect a large distributed system on a whiteboard; a lot of people applying for a senior software engineer position lack the breadth of knowledge and the consideration required.
And after that you'd speak to CTO, where maybe you can say or do something so silly to be rejected; this happens very rarely.