mine have lots of hammocks and tunnels and snugs (i got them from
hammockyhammocks.co.uk - the lady who makes them has chinchillas herself; i've started sewing last year so i make my own as well now)
fleece is deemed safe for chins but from my experience, most shop sold hammocks etc are either nylon fabric or furry stuff - i wouldn't go for either of them. if you do find a fleece (including straps) one, you may still find that your particular chin just likes to chew
some do and some don't.
my lot love the fleece stuff and don't chew it. i have one snug that's lined with polar fleece and my rescue girl sometimes has that one (she has the tunnel at the moment) - she's had teeth problems so i knew she'd definitely be ok with it but now she's eating wooden chews, i'll just have to make sure she's not eating it when i put it in.
so basically, you're looking for fleece and need to be vigilant to see if there are any signs of damage
if you can't find any where you live, have a look at hammocky hammocks. the prices are very reasonable, the products are fab and she ships internationally. the only 'thing' is that because it's just her doing this and all proceeds go to chin/piggy charities, it can sometimes take a while - the waiting times are on the website. however, i do think it's worth the wait and i know she's really good at getting stuff out quick if you have an emergency (like when you get a rescue unexpectedly!)