I think it's generally accepted that they like floor space more - from what I've read on other forums. Levels
can be used to increase floor space but this needs to be done carefully (ie, large, un-cramped levels and ensuring they overlap to prevent falls from top to bottom of a tall cage).
Is it not possible, like Basia says, for you to get some firm mesh and make either a window in the lid or side of your bin? I'd have thought that would be more secure than an open cage like that tall Crittertrail you mentioned.
It's not that hard - I'm not very DIY savvy and I recently made my travel boxes which are identical to a bin cage except smaller. The lids lock in place and they seem very secure:
You'd need to screw the mesh in place rather than zip-tie it for a permanent residence, I think, but other than that it would be the same - each one took less than two hours to make.