As you say, umbrella companies have considerable costs of their own which they have to recover, as well as trying to make a profit. You could set yourself up as a sole trader or alternatively create a Ltd company. Either way, you'll be responsible for your own tax and National Insurance arrangements, so you'll probably want to employ an accountant to sort it all out for you - more expense.
Why not become an employee and let someone else worry about it all? I started as a contractor but was only working for one school; eventually they said to me that that didn't want to be exposed to a possible difficulty from HMRC and took me on as a salaried instructor. The major benefit to being a contractor is that you can claim travel to/from work but if you only work for one school at a time, HMRC might catch up with you. Do a search on pprune and find plenty on the subject!
TheOddOne