You've identified a weakness which is similar for a lot of small business owners - you understand *your* business, but not necessarily where you can fit in to others.
A few thoughts that I've had on reading your post:
1. How is the schools IT market developing? With all UK schools transitioning into Multi Academy Trusts in the near future, I'd have thought that IT provision would be an obvious choice for centralised resource sharing and cost saving. You're also targeting a market which is not known for it's excessive budgets - do you have any other verticals in your client base?
2. You'll have heard this before, but your network is the best source of referrals. Be the company that's an instant recommendation for IT problems. In the past year, all 40 of our new clients were referred to us by someone who already trusted us.
3. Last year I did the Help To Grow Management scheme : https:/
Delivered by a local business school, including ongoing business mentoring, it's left me using a lot of what I used to call "bullshit management speak" but has been genuinely helpful and a big part in boosting our turnover and profits. It's 90% funded by the government - cost to you is £750.
If that's not suitable for you, there are similar schemes run by local small business groups which might help you.