The best way is Linked In. Reach out to people from your known circle with a referral request.

The way you approach shouldn’t want them to skip your message: You can start the message by greeting them and telling them your current proficiency and the reason to join the company, by sending your resume.

Hope this helps.

Find a 1st degree connection/ friend who works in a big firm or a connection who knows someone working in a big firm. It's always best to go through an acquaintance, college alumni, ex-managers etc

Search on linkedin for the org you are looking for. Find 1st and 2nd degree connections. Find mutual friends. Tell your friends/acquaintances what you are looking for. It's even better if you have a JD handy , you can share the JD and ask some time to refer you against that. Share context on why you are looking out and why you feel you may be suitable for the role.