Hiring a veteran is a risk.
Think of this from a recruiters perspective. Hiring a veteran can be very risky to someone that may not know much about military service. Veterans come in all shapes and sizes, and there is little to predict whether they will be able to add immediate value to your organization, especially in a demanding role like Product Management. I propose using my "3 P's" to showcase your strengths and decrease the perceived risk in hiring a veteran like yourself.
If you're looking to start your post-Military career as a Product Manager by joining an elite Fortune 500 tech company in the valley, or even looking at a startup, you must be able to leverage these three things: Portfolio, Pedigree, and People. Done well, you'll be able to make a convincing argument for that dream PM job.
Portfolio: What have you delivered? Can you show a track record of success? Can you demonstrate that you're the type of person that can create a vision for something and execute it to completion? If so, showcase these accomplishments proudly and make a connection to how you can carry those skills to delivering software as a Product Manager. Non-military examples are better here, so if you've never built something from scratch, get started with something easy! Build a website, create a store and sell something on Etsy, or code a simple game. Show the interviewer/recruiter that you have experience thinking through product problems.
Pedigree: What people/institutions/brands are you associated with? Are they a credible source with a high brand value? The more quality people and institutions put their name behind you, the less risky you seem. This is most clearly seen with what universities you got your undergrad or graduate degree from. Military units can also convey pedigree. Being associated with a high brand value unit or MOS can show that you've met the rigors of that organization and can also do so as a PM.
People: Who is willing to vouch for you? Without a doubt the best way to land a job in any function or industry is to personally know a hiring manager. Absent that, you may know someone that can refer you to a specific position you're looking for. Develop strong overlapping networks that you can use to your advantage. Make sure that you keep those people updated with your job status, preferences, and skill sets. Use your network to source jobs for you.
These 3 P's to landing a PM job are absolutely critical especially if you don't come from a technical background. While some companies, roles, and hiring managers could be looking for a technical product manager that may require extensive experience or technical degrees, there are many other product management roles that don't require the technical background. In these cases, the 3 P's are especially important make you competitive against the other non-veteran applicants.