Not sure if you could do a whole day in the park. (You’d have to be pretty fit!). The red loop is definitely worth doing, maybe extend the ride into the southdowns/Meon Valley area?
There’s a nice Meon Valley loop which starts at Butser (other side of the A3(M) from QE), then head back to QE for a few more loops? I’ll send you details if you want.
There’s other riding out of the park on the SDW but quite a lot of road to get to the good stuff.
Or there’s all the cheeky stuff, of course, which someone will no doubt come along and tell you about!

Here are some longer options: http://www.qecptrailcollective.co.uk/trails/xc-routes-in-the-area/

Any linked with a lap of the red trail would be a good days ride.

“Bridleway Medium Route 2” on Scott’s page is the one I was referring to.
Couple of nice sections on that route.