[bentona]

Some advice based solely on my experience, YMMV etc.

Re: Mentors - reach out to someone who you enjoyed working with in a previous role, looked up to, and would enjoy working with again. Just ask them to have a quick chat, and have specific questions about your job hunt ready - ask them to share their approach to a job search, share what you've done, and ask if they have any advice for improvement. If the chat goes well, you can ask if it's alright if you reach out again. You don't necessarily need a full "mentor", just some guidance.

Re: Referrals - This is not strictly about typical "referrals", but if you can have a chat with a recruiter, you often can similarly cut through initial screens / red tape.

I have had a ton of luck reaching out directly to in-house recruiters for roles I'm interested in on LinkedIn. They're often constantly on LinkedIn, frequently doing outbound, and IME they're often happy that someone reached out - and this can be more effective than spending an hour hand-crafting a cover letter. Keep the message to a couple sentences, share why you're interested in the company, and again, ask to have a quick chat.