I was having the same issue until I realized I have to look for the expedition logo on the map to get the “white chests.” The gear chests and currency chests usually contain very little.
Start off surveying the map for the remnants that give “excavated chests contain 20% more items.” Try to hit several of these early in your chain, then get as many white chests (expedition logo on map) as possible. Every other chest is not worth going for. Prioritize quantity of excavated chest modifiers, then hit as many of those white chests as possible, then if you still can, try to hit an underground area that may have another white chest or two in it.
I went from getting 20-40 splinters (doing it the wrong way not knowing what I was looking for), to getting 100+ in a single logbook.