Hey Brittany! This is always the conundrum, isn’t it. I’ve gone through about 1,000 different hay dispenser configurations before finding something that seems to be working okay. I should think that it’s more natural for rabbits to “forage” around near the ground, as you observed. Theodore had absolutely no idea what to do with a traditional hay rack, and thus it was ditched immediately.
Right now, I have a small plastic basket attached using some plastic clips (such as the kind that come with hay bags or even around some bread bags that keep the bags closed) that “pours” into the litterbox. 

 Here’s a photo, (disclaimer: you’ll notice that there seems to be a lot of hay under the basket. Don’t let this fool you — that’s just because I was messing around with different configurations and spilled hay everywhere in the process.): 

I find this set-up keeps the hay to one spot, mostly the basket and litterbox. It’s pretty much inevitable to get some spill over, but this so far has been very clean. Theodore is prompted to hop into the litterbox to get to the lower end of the basket, which reinforces his litterbox skills, and then he can forage and munch away, and then the excess hay falls right into the box. Very neat.

I put the hay on a cotton quilted placemat on the floor.

I have Porky’s hay in a wire basket hanging just about at litterbox height. It is not in the litter, so stays clean, but it is very low so he isn’t reaching up for it. Also since it is a wire basket it is very very open and bits fall out very easily, so he can get those off the floor if he wants.