Hi,
Have have 2 chinchilla's & I clean their cages out daily!
I put their hay in enamel bowls & they always dig around in it for the 'best bits' so its just normal for them to do this.
I'm afraid you will just have to remove as much as you can. I have sometimes left bits of it lying around but they don't seem to bother with it - they just eat the hay from their bowls.
I have found a very good website about chinnies -
www.chinchillas-unlimited.info
The forum is very informative & everyone is friendly