Ah, yes alfalfa pellets, decomposing hay, all the same thing really...
When we break down greens we add nitrogen. Nitrogen is bacterial food, and bacteria add sticky humus and more nitrogen to the soil. Shade and moisture allow for habitat (hot and dry don't).
I bet bunnies add a few good things in the biological mix, and they
are cute.But they use some of the nitrogen in the alfalfa ( which is very high as far as organic goes)...
Adding fungal side is harder, but builds soils faster. Try shredded paper under your hay, maybe a layer of leaf mold in between (there are SOME trees in Central Texas, right?

) And vetch DOES add nitrogen; but getting rid of it later might be tough. I'd compost unless you are talking about acreage...
HG