Hi there & welcome to the forum,
when mine were out in the shed in their hutch i used to use a thick layer of newspaper about 20 or so pages deep, all along the bottom of the hutch, if you layer it slightly overlapping, i would then use about an inch thick or more of short cut, shredded paper & then topped it off with a couple of big heaps of hay. (where i work they have cheap looking recycled paper shredded& not the bright white stuff, but it's more absorbent so better for the piggies & i can have it for free
)
its easy to spot clean with the short shredded paper & then when you do a proper clean just roll the under paper up towards you & it all rolls into a big sausage that can be easily put into the bin (easy lazy way
)
if you wanted to use that method in the open area of your hutch & then use the fleeces on an absorbent layer in the bedroom area, you could save on the fleece & it would keep their bedroom warm & cosy
also for the open end if you could find a large piece of polythene or tarpaulin to go over the mesh just to stop the wet getting blown in or too draughty it will help too. (make sure there's enough ventilation though)
some supermarkets or businesses have pallets covered with thick polythene (not cling film) that they usually just dispose of, try asking if you could have some. cheaper than buying it & recycling/ecofriendly too
only ask for polythene off of piggy friendly pallets (food or fabric etc pallets, not chemical pallets in case anything has got from the pallet to the polythene).
i wouldn't use cotton wool as it would go lumpy when being washed, if you can get some wadding or just fold over old towels to use as filling to your fleeces.
But hay is essential at all times for their dietary requirements, they must have it to provide the fibre they need & also to wear down their teeth & help keep dental issues at bay
xx