actually i'm guessing westy doesn't reallly know the area too well - i can suggest a bit of a weird loops that should be pretty fun, although not stricktly basingstoke it does start and finish there
head from b-stoke over to hook on the london road, throught hartley whitey on the odiham road (very nice stretch here), up bracknell lane, back on odiham road to riseley, spin round at roundabout just outside it and back down odiam road to hook, drink at the leather bottle and the m4 back to b-stoke (looks something like this:
http://www.gmap-pedometer.com/?r=234362)
oh and it would be a shame not to do the churchill way(center of b-stoke)->m4 run, thats great fun
edit: looking at it, thats about 30miles worth of stuff, mainly national limited B roads, should be albe to do it in about 20min at a guess, so could extend upto Reading, across towards newbury and back down the a4-a340 which is a really nice road, IF theres no traffic infront - zero overtaking points