My two cents.

I don't think it is wrong to be "that guy" who lands a job offer from others. This industry is 50% networking, and 50% experience. It's probably 90%/10% at some companies.

1. Use people that you know well or at least you feel confident talking to HR staff in case they contact the individual directly.
2. Use people that know you well. Nothing discredits a source if that person knows nothing about you (your regular life, family, kids, etc.etc.), your background, or significant achievements mentioned in the LOR letter.
3. Keep in touch via professional organizations (Union, safety programs, flying clubs, fraternal organizations etc.etc.)
4. If not, keep contact via social media. This is huge ... you will run into many others that you've lost contact with. Facebook was the single greatest tool that I used to network when landing my dream job.
5. Share the love. Whenever I flew with or worked with someone that impressed me, I always relayed to that person that it was okay for them to contact me if they had any questions or needed any help professionally. In fact, I encouraged them to do so.
6. If you don't have any connections, you are effing up. You need to go back into your list of pilots and instructors and make contact. Become professional friends. Become personal friends if possible. Find them on Facebook. Run into them at the airport. Do what it takes.
7. Don't be shy. If you know someone that could be used for a recommendation, don't lie about why you are contacting them. Ask questions about the company. Find out the good and bad. And be upfront, ask them if they would be able to sponsor, recommend, write a LOR --whatever the case. If they say yes, don't make the person recall from memory stuff about you. Send that person your resume, maybe a letter detailing your life and family, and what you've been up to for the past whatever years. Also, it may be helpful for you to write your own LOR, with specific "there we were stories", reminding your contact about situations you feel might be good to highlight. The writer can then decide to edit your letter, or write their own. Find out what they want. But at least offer the information so they aren't recalling information about you from memory.

Research Emerald Coast Interview Consultants ... they give a lesson on this and have much better information and tips than I can share on a Super Bowl Sunday!