Yeap - cold calling is very tedious and the conversion rate is quite low. As Jeff S suggests, targeting an association would simplify marketing the solution. A simple google search yielded one national organization for high school athletics coaches, NFHS, and numerous state-level football coaches associations. You could always target those associations as well as affiliated organizations and seek to advertise in their publications or websites. However, I'm wondering why and how you arrived at the price point you list. A $20 product seems quite low for a product (I don't know what it is or the value it provides, so I apologize in advance if I'm mistaken).
A quick and dirty market analysis might go like this: The Department of Education suggests there are approx 24,300 public secondary schools in the US. Let's assume there's an equal number of private secondary schools, all with football programs. At $20 per school (assuming the price is per-school or per-school per year), the total addressable market value is $972,000 (24,300*2*$20). If your pricing model works on a recurring revenue basis, then you can earn about $972,000 in revenue per year, assuming all schools renew. Concurrently, if you can sell multiple products into one school, the addressable market could be much greater in value.
With that said, cold calling might give you a conversion rate of 15%. If that's the case, then assuming you call all schools and you convert 15% of those calls (one unit per school), you can earn $145,800, which isn't a lot of money. If the $20 is a product, I'd be looking at the gross and net margins. Similarly, if the $20 is a solution, I'd also be looking at the margins I could earn. In the end, your marketing budget should be sufficient to cover advertising expenses and take into account the average cost to make a sale. The $20 product may end up turning a very low margin or even cost you money to make and sell.
Lastly, if your selling a product, you can always sell direct or through an established channel. I don't know the specifics but I'd guess high school athletics departments buy products from a vendor that carries a bunch of products. Pitching and selling your products for inclusion into such a vendor's catalog would help you reach a wider audience much easier.
I'm digressing from your original question but I'm trying to understand the underlying fundamentals of the business model and the budget you may have to market such a product. I've seen too many people ignore pricing models and jump head-first into a venture that ended up costing more money than was ever made.