Factory paint is usually pretty hard (as long as it's not been re-finished in the past - it doesn't sound like it), which is good news as what you need is a 'power polish'. T-Cut is all good, but the shade you bring up can vary throughout the car depending on your arm strength! The best results would come from the power polish (basically using a specialist electric buffer.) I reckon if you went to a bodyshop and asked them, they'd buzz over it for a few quid. They basically use T-cut (which has a cutting compound in it), but better (usually 3M gear) in two different grades (coarse & fine) with two different mops materials too, and pull the very thin top layer of original paint off, exposing the true colour underneath, they buff it from there with a decent polish. You could do the polish op yourself by hand once.
It'll look like a different car