Yes, hay will work. I don't know about parting a rolled bale and making a deposit. Better to build a base of hay and then alternate between your deposits and layers of more hay.

In my experience you can use any loose or fibrous vegetable material as long as it is dry. Hay, straw (yes, there is a difference), dry leaves raked off the lawn, lawn clippings if they are left out in the summer sun and thoroughly dried, dried sphagnum moss. A bout the only thing that I haven't seen work well is pine straw. Even dried it doesn't decompose rapidly enough for composting toilet that gets used more than once a fortnight.

The dry, compressed peat moss sold in blocks at any garden center is what I store for emergency use. I like it because it is tightly compressed, dead-dry, and it comes in a nice cube that is sealed in plastic. It doesn't attract insects or rodents and seems to store forever.

The key is to keep the composting stack, a base of dry material and alternating layers after that, dry. The only moisture should come from the manure you add. Absolutely no urine, water, or, especially, cleaning or deodorizing products. You also need good ventilation. If your compost stack is sealed in a bag or container it will turn foul and it will smell worse than the manure you're trying handle. Odor is controlled by keeping oxygen available to the bacteria until your manure dries. Dry and largely decomposed the waste doesn't have any significant smell and can be stirred into the soil around your shrubbery at any convenient time.