I have a cheapo solution for a deep compartment dry fly box- go to a drug store and find the section for pill boxes.
Some of the boxes will be multi-compartment, to hold 7, 14, 28 days worth of pill assortments. Some of them work better than others. I avoid boxes that are too big, that use a common lid instead of separate latched compartments, that have compartments that are too small, that have shaded color compartments that you can't see into, or that use cheap latches that might open without much pressure and accidentally spill your files.
The ones I prefer have clear and rectangular compartments, with seven separately opening ones in a single row, use a nice size plastic press latch (about thumb width) on each separate compartment, and they're deep and wide enough to hold at least one #6 Royal Wulff without squashing the hackles. They're about $7-$8 per box. Better than any spring compartment box or foam box I've ever used. Those go into my shoulder bag.
I also like to carry a smaller multi-compartment pill box or two, in a size that fits into a shirt pocket. The compartments on the ones I've found in that size aren't as large, but they'll fit dry flies up to about size 12. Important to find one with good snap latches. I pre-select my favorites for a given day and put them in the shirt pocket box. Ideally, that should be the only fly box I need to use on the stream. In theory, anyway.
Always a good idea to store the flies you've changed out and snipped off while fishing in unused compartments, and be sure to open the boxes up and let them air out to dry when you're done...
Also important to carry the tools to complete the task: scissors, needle nose pliers to mash barbs, hook file, great big ole obvious needle threader, and bodkin to open up obstructed hook eyes (although the point of another hook usually works pretty well).
If I fished tricos and other micros, I'd probably want one of those C&R midge boxes with the set of threaders, to have those invisible things all set up to tie on to the tippet. But for anything down to size #20, my usual boxes work fine.