I have used a utility/equipment trailer to haul gravel and dirt. I had 2x10"'s around the front and both sides and held it it place with 2x4's through the stake pockets. If your wood floor is not very tight, I have used a either a sheet of heavy plastic or a cheap blue tarp placed on the floor to keep the material from falling through the cracks. (You will be sacrificing the tarp/plastic when you unload.) Now drive the CUT onto the rear ramps to unload. You will be able to remove 90% of the load fairly easily with this method. Try to unload using the FEL bucket just raised up over the tarp/plastic as you do this. You may be able to reuse the tarp/plastic once or twice, if you are careful.
Last year I hauled by tractor to BIL and removed some dirt near his house to get better drainage. Hauled my tractor and few attachments on my 24', 14,000 lb tilt bed trailer. Used the tilt bed to haul the dirt away. It worked really well. The floor is very tight on that trailer and the tilt bed made it very easy to unload with the tractor. I could haul about 5-6 yards with each load but was probably overloaded. Driving up ramps and using a FEL takes a little more skill but worked ok. Much easier and faster than unloading by a shovel. A utility trailer with low wood sides can hold a lot more material than the trailer is rated for, so don't over load the trailer.