Hi Jay.
I'm exactly the same as you, 23 y/o (24 in Sept.), 3 years no claims, I actually have 3 points on my licence that come off Nov. this year (Insurance Co. didn't actually care about these anyway). I'm with Admiral and started back in March for £930 roughly .
I know Admiral, Bell and a few others are behind the scenes the same company, but try them online on their OWN website, you get like 10% off just for doing it through their online quote service!