"…do you fill the hole up with dirt (to the bottom of the rim) so your trap is more solid…"
You are missing one of the prime advantages in using this system. Having a hollow spot underneath the middle of the trap keeps the trap from becoming "high centered" on the chain/swivels or whatever hardware you have underneath there. If you put too much dirt under the trap, it will become high centered and will be less
As 920 said above, ordinarily a lot of dirt will fall back into the hole when you pound down the rim. If not, don't be afraid to throw a handful or two under the trap before you bed it. Just don't use so much that the hardware will not pack down into it.
That said, if you are forced to use your dry dirt, or some other non-ambient trap covering, then you may want to go ahead and use a pan cover. I use a coffee filter because it will fit under both jaws and effectively seal off the bottom of the trap, saving dirt from falling down into the hole.