For units that don't offer rushing, it can be a little harder to stand out. That said, it just puts it more on you to have a solid, squared-away packet that stands out. Have a good picture of your face. Even if they ask for full length, nothing says you can't have another of your mug to help make you recognizable if/when you do get invited to a meet and greet/rush weekend/interview. Honestly, I used my firehouse headshot with my helmet on. Cheesy, for sure, but it stood out and made me more recognizable as the old geezer fireman trying for a spot.
Make sure your resume looks professional. Have business-minded friends or colleagues look at it, or feel free to send it to me, if you want random stranger feedback on it. Include unofficial copies of your transcripts, any flying stuff you have, and your LoRs in the packet. Make sure it's all in one PDF and not multiple files.
Spend some serious time on your cover letter and, like your resume, have it proofread by friends/coworkers (or send it to me). It should have a little of everything; who you are, why you want to fly, why you want to fly for that specific squadron, and what kind of asset you want to be to your squadron after training. Keep it to one page.
Lastly, make contact with the hiring PoCs through phone or email (or both). They're getting blasted with packets from a bunch of people they don't know (who are probably blasting packets out to lots of squadrons themselves), so all you are is numbers and letters unless you make contact. Try to speak with other pilots to ask questions, express interest, and (without sounding like a bragging douche) talking a little about yourself to have them get to know you a little. The pilots will talk and, if they know who you are, I'm sure they can help the hiring board with input on who to invite for interviews. It'll also make it less awkward when you do show up if you have spoken with a few pilots in the squadron to strike up conversations.
Bottom line, you've gotta create your own luck and that's by putting in most of the heavy lifting yourself. You don't want to be psycho stalker level of bugging them, but you want to make sure you're making solid contacts and showing you're interested, if you can. The squeaky wheel getting the grease and all. Just sending your packet out and hoping the numbers and letters on the pages are enough to get you a job offer might not be enough when you have plenty of other solid candidates; you've got to have personality do some of the work.
Good luck!