If you choose "Path for Rare-Treasure Hunting" then you will get more rare items such as Fossils (Only in USUM) , Big nuggets , Comet Shards , Pearl Strings and Golden bottle caps. However you have to develop your "Isle Aphun" to phrase 3 and you need to make sure you have at least 60 Pokemon in your PC Boxes.
It takes about a day to finish cave hunting (expedition) but you can speed it up by putting some Pokebeans in the box.
Also you can buy a Jaw Fossil at Konikoni City (Olivia's jewelry shop) but it's only avaliable in Ultra Sun.