You have a fundamental misunderstanding about what a PR company does. You're thinking about an advertising agency. They're not the same thing.
It's like asking someone on HN "Tell me how to hire a good person." I can't hand-hold someone through the internet.
Any meaningful connection is going to be too expensive for someone like GP, I am afraid.
> Tech people are notoriously bad at public relations
Please avoid (false) generalizations.
This is a false generalization. You don't have to choose a big-name New York PR agency. There are thousands of other shops that are quite affordable. Some are just one or two people, and can be surprisingly effective if you pick the right one.
In other words, even if the PR firm is small and one of the best and cheap, they cannot really influence much whatever their contacts offer. They may have some leverage, but the price is the price...
It's not about "connections." It's about doing business the old-fashioned way: advertising, and hiring a public relations agency.
Advertising is self-explanatory. But for some reason a lot of tech companies don't hire PR firms. They like to cheap out and do PR in-house, or they simply never think of it.
It's the PR firm's job to have the "media connections" you so desperately crave. Tech people are notoriously bad at public relations, so it's perfectly logical to farm this out to people who specialize in just this sort of thing. There are even boutique PR agencies for various industries, including tech. But in your case, you should have hired one of the several hundred that specialize in education.
You could have been the next Oregon Trail!