I think I have a little bit of expertise in this arena.
I am a Trustee of my State's Boy's State as well as the coordinator for my County and finally, I am a staffer when we actually DO Boy's State in June.
A few notes: We have one of the largest programs in the country - over 900 delegates and we get frequent staff "riders" from other Boy's State programs to try to capture how we do it.
My county has 21 Legion posts and sends about 80 boys each year. Not all posts participate but most do and we also sponsor a few from the County itself (that I handle). Our state considers a high school to be involved/participating if they have sent a boy within the last five years and at this time, I can tell you that every high school in my county is covered even ones that do not have nearby Legion Posts. This includes both public and private schools. The sticky point over the past few years seems to be kids who live in our county and attend private school in another county or vice versa but we have handled that as well. One thing we also have here is the ability for "Self Sponsor" where kids from schools without Legion Support can still attend on their parent's dime which this year will be $275 for the week.
Now, to the OP's question: The Boys State "year" has not really begun in our state yet. I can't say that your state is the same but it is likely to be similar. In the next couple of months, Posts will reserve slots at Boys State and then begin to reach out to "their" High Schools. In my experience, February/March are the big months when things start moving and our state website starts collecting completed applications from posts in April but the bulk come in May. We have a wide variety of large/small/rural/etc schools in this county but almost all of them cooperate with the program most years. I personally handle a few schools that would otherwise drop through the cracks as well as provide assistance to other schools/Legion posts.
When I find a guidance counselor who is unaware of the program, I go to the Director of Guidance and they are in most cases aware of the program and willing to assist. If you try to reach out through your local post, first of all, don't just talk to the person who answers the phone who very well could be a bartender or maintenance person and totally unaware of the program. This is a question for the Boys State coordinator but since they might be hard to identify, reach out to the Post Commander who can often be identified from a post website, google search or social media. If you do call the post and get the random phone picker-upper, ask for the Post Commander and get in touch with them. If you are still unsuccessful, do the same contact attempt to the County or District commander who can usually be identified by the same means that I indicated for the Post Commander.
For example, a few of our county posts have websites (a third to a half are on facebook) but all the posts and the current commanders are listed on our County Legion website as well as our County Leadership and there is contact info for these people. We also have a state legion website which identifies the County Leaders for each County. Our Boys State Website is probably the best resource in our case:
www.aljbs.org as it includes a lot of information about applying, lists all of the county coordinators (with contact info), lists every high school in the state and if they have participated in the past 5 yrs and much more.