Lots of things to play with, how is the ground under the antenna? rocky sandy dry. clay moist etc.
Radials are jsut one part of the equation, that will make the antenna more efficient.
The far field will be just as important as well, ( 10 - 100 times the wavelength) this will determine the angle you will get with the antenna length.

I live on young seaclay here, very good ground for H.F. the next better one is pure seawater.
That is how i get away with a mediocre L antenna and a not perfect radial field on the edge of the city and still can DX on 160 and 80 wityh the L.

Consider an L, horizontal/vertical polarisation, will give you NVIS contacts as well DX.
for 80, 10 meters vertical, 10 meters horizontal even if that is sloped down to 10 feet above the ground.

No need for an tophat, the horizontal section is the tophat and actually radiates, the tophat does not radiate...it is just the missing C for the too short antenna.
Currents in the wire sections of an tophat cancel each other.
If you use an tuner at the feedpoint you can use it from 80 and up as allband antenna, my L works from 160 - 10 meters.

I tried, and stepped away from using baluns/UNUN's at the bottom to feed them for multiband.
I know the 43 feet and 30 feet verticals are used that way, but i found the losses too big, a rasonable tuner fed at the bottom of the antenna has lots more efficiency.

Even an isolated wire thrown over a tree will work, an inverted U antenna ;) with 100 watts i never set the French countryside on fire with that :)

http://www.dxzone.com/cgi-bin/dir/jump2.cgi?ID=7479 forget the balun, and put tuner at the bottom.