Hi there,
Regardless of your situation, you should absolutely try to network your way in. This is all the more true when you're coming from a low-odds background. Now, in terms of improving your odds + obtaining referrals, I have the following advice:
How do you improve your screening chances?
1) Gain experience (internships, bootcamps, classes, stratups, extracirriculars, etc).
2) Network - speak to multiple people across offices and aim to get a referral
3) Optimizing your resume and cover letter - make sure your formatting is sharp/consistent, key words/experiences pop, and your experience bullets are properly written.
In regards to networking tips:
a) Reach out to people in your network
b) Reach out to people once removed from your network
c) Reach out to people with a similar backgorund to you (i.e. same alma mater, same historically underrepresented demographic i.e. gender, orientation, ethnicity, etc., same career switch, etc.)
d) Tailor a message to them specifically both showing interest in them and their journey and demonstrating that you have done your research and could be a valuable hire
e) Play "tag" across calls you get so that you can work your way towards the company/office/role you want
f) Never directly ask for a referral, but "hint" at needing one (this is nuanced and important...happy to talk through wording)