We have the same hay basket, and purchased a piece of netting to cover it (think the original purpose was for hockey or soccer goals) which was large enough to fall into the empty hay basket. This gave enough slack that the horses could continue eating as the hay was consumed, and the level went down.
We paid the slight upcharge to have the net bound around the edges, and used giant versions of those elastic loops with balls that are hair accessories to pull long hair back into a ponytail. Looped them through the netting, and around the metal pipe of the hay basket.
At one corner of the net (it was square), we put a double-ended snap that we snapped onto a stout nylon dog collar fastened to the hay basket (to the metal ring that normally holds dog tags). This snap was easy to undo, allowing the net to be lifted and hay put into the basket, without undoing the elastic loops holding the rest of the net in place. When we did want to remove the net entirely, for instance to clean the basket after rain, it wasn’t difficult to remove and re-install the elastic loops.
Worked quite well for a long time - and the neighbor who fed for us occasionally didn’t mind it – but we eventually quit bothering with the net, as we didn’t think it slowed the horses down all that much.
I can visualize a version of what we did done with those velcro loops used for cord management, in place of the elastic loops.