In nature rabbits eat more leave-like greens than grass (what hay is produced from). So you could use any rabbits' taste fitting leaves, like European acorn, dandelion, European oak, apple, plum, daisy, strawberry, raspberry, peach, carrot, garden radish, cabbage (if you do not feed grains), hazel, European chestnut, European linden tree...
There are lots of possibilities! From all plants mentioned above, I speak about their leaves. Most of the time, the fruits are too rich in energy to be a base for good food. Some could be given as treats, but one needs to have in mind, that these fruits in the nature are only available 2-3 months per year.
(If you think about the "European" in front of the trees, this is because the domestic rabbits descend from the European rabbit, so the European kind of tress and bushes are these ones, they could eat safely. I do not know whether it is the same in case of American or other kinds of the same family, so it may be a risk to feed them.)
PS: In nature the rabbits would always choose fresh leaves instead of dried ones or hay. The most time of the year fresh leaves are available, so pet owners have no need to force their rabbits to eat hay, when they eat enough fresh leave-like greens instead.