Working under an umbrella company means you are effectively an employee doing PAYE. Effectively, you will pay the most tax via this method because the umbrella company is doing all of the legwork of dealing with your pay and tax.
I've been doing IT contractor work but have set up my own limited company so pay quite a bit less tax (legally, of course) but also had to get (and pay) an accountant, setup the company with companies house, get VAT registered, but there are quite a lot of tax breaks and you can take a lot of expenses and there's not that much more to it.
I'm not considered self employed however, but instead am the sole employee of my limited company so mostly pay myself dividends but also a little PAYE and can do salary sacrifice and the like for pensions, etc.
I really don't know how it works in the USA, if you similarly setup an LLC or something like that.