Paper backgrounds would generally always have to go on the outside of cages(even so called soy based ink can have artificial dyes which can cause health problems over time so are best avoided in the cage)no matter what.
A natural option would be a particularly knotted natural grape vine piece(as with any product I'd advise not using anything from China,but it shouldn't be too hard to find USA sourced grape pieces).Make sure they're not too tall!
Hamster safe natural wood sealant and good natural paint is a good way to customize things that are already natural and avoid using paints with harsh ingredients(including artificial dyes and other things,as practically all conventional paints will have).Pure beeswax,sourced properly,is a wonderful example for wood sealant.
Certain things could be used within the cage to add depth
A good example would be oat sprays(make sure there are no additives,Brown's oat sprays are a suitable option as they're additive free and not from China).
https://www.chewy.com/browns-tropica...E&gclsrc=aw.ds
Brown's also has wheat sprays,and while many of their products are quite low quality the previous product and this aren't bad(I'd avoid their corn cobs and any other corn cobs being in the cage though as corn cob tends to mold far too easily to be worth risking):
https://www.chewy.com/browns-tropica...ural/dp/124735